{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-16T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":1046,"title":"Add two numbers","description":"Calculate the sum of two numbers.\r\n\r\nExample\r\n\r\n input  = [2 3]\r\n output =  5","description_html":"\u003cp\u003eCalculate the sum of two numbers.\u003c/p\u003e\u003cp\u003eExample\u003c/p\u003e\u003cpre\u003e input  = [2 3]\r\n output =  5\u003c/pre\u003e","function_template":"function y = addup(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [2 3];\r\ny_correct = 5;\r\nassert(isequal(addup(x),y_correct))\r\n\r\n%%\r\nx = [-3 3];\r\ny_correct = 0;\r\nassert(isequal(addup(x),y_correct))\r\n\r\n%%\r\nx = [2.1 48];\r\ny_correct = 50.1;\r\nassert(isequal(addup(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":11,"comments_count":3,"created_by":6303,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":1487,"test_suite_updated_at":"2013-07-11T15:49:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-11-21T13:35:24.000Z","updated_at":"2026-04-13T21:34:31.000Z","published_at":"2012-11-21T13:35:30.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the sum of two numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ input  = [2 3]\\n output =  5]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":42450,"title":"Summation of array","description":"Given an array, Find the sum of all of the elements in it\r\n\r\nExamples:\r\n\r\nInput x = [1 2 3 5; 4 5 6 7];\r\n\r\n\r\nOutput y is 33\r\n","description_html":"\u003cp\u003eGiven an array, Find the sum of all of the elements in it\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cp\u003eInput x = [1 2 3 5; 4 5 6 7];\u003c/p\u003e\u003cp\u003eOutput y is 33\u003c/p\u003e","function_template":"function y = sum_array(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [9 7 10;\r\n    10 1 10;\r\n     2 3  2;\r\n    10 6 10];\r\ny_correct = 80;\r\nassert(isequal(sum_array(x),y_correct))\r\n\r\n%%\r\nx = [4 8 7 2 9;\r\n     6 9 8 3 6;\r\n     7 6 5 5 8;\r\n     5 4 1 6 7;\r\n     9 2 3 5 5];\r\ny_correct = 140;\r\nassert(isequal(sum_array(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":45298,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":374,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-07-07T20:06:58.000Z","updated_at":"2026-04-12T21:25:36.000Z","published_at":"2015-07-07T20:09:47.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven an array, Find the sum of all of the elements in it\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInput x = [1 2 3 5; 4 5 6 7];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput y is 33\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":43100,"title":"Add Me To Death (Sum sum sum)","description":"Given certain vector,  sum of the input vector, get the sum of 2 adjacent numbers in the input, and sum of the output vector (which is the sum of the 2  adjacent nos)\r\nSum sum sum :)","description_html":"\u003cp\u003eGiven certain vector,  sum of the input vector, get the sum of 2 adjacent numbers in the input, and sum of the output vector (which is the sum of the 2  adjacent nos)\r\nSum sum sum :)\u003c/p\u003e","function_template":"function [a b c] = addMeToDeath(x)\r\n  \r\nend","test_suite":"%%\r\nx = [1 2 3 4 5];\r\nsum_input  = 15;\r\nsum_output = 24;\r\nvec_sum_adjacent = [3 5 7 9]\r\n[a b c] = addMeToDeath(x)\r\nassert(isequal(a,sum_input))\r\nassert(isequal(b,sum_output))\r\nassert(isequal(c,vec_sum_adjacent))\r\n%%\r\nx = [7 8 9];\r\nsum_input  = 24;\r\nsum_output = 32;\r\nvec_sum_adjacent = [15 17]\r\n[a b c] = addMeToDeath(x)\r\nassert(isequal(a,sum_input))\r\nassert(isequal(b,sum_output))\r\nassert(isequal(c,vec_sum_adjacent))","published":true,"deleted":false,"likes_count":8,"comments_count":0,"created_by":13865,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":74,"test_suite_updated_at":"2016-10-29T17:05:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-06T07:20:13.000Z","updated_at":"2026-04-09T21:23:28.000Z","published_at":"2016-10-06T07:20:13.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven certain vector, sum of the input vector, get the sum of 2 adjacent numbers in the input, and sum of the output vector (which is the sum of the 2 adjacent nos) Sum sum sum :)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":730,"title":"How many trades represent all the profit?","description":"Given a list of results from trades made:\r\n\r\n[1 3 -4 2 -1 2 3]\r\n\r\nWe can add them up to see this series of trades made a profit of 6.  \r\n\r\nSince the two biggest trades sum to that, we can say that all the profit came from two trades.\r\n\r\nWhat is the minimum number of trades that represent the total profit (possibly exceeding it).  All trade lists will represent profitable series.","description_html":"\u003cp\u003eGiven a list of results from trades made:\u003c/p\u003e\u003cp\u003e[1 3 -4 2 -1 2 3]\u003c/p\u003e\u003cp\u003eWe can add them up to see this series of trades made a profit of 6.\u003c/p\u003e\u003cp\u003eSince the two biggest trades sum to that, we can say that all the profit came from two trades.\u003c/p\u003e\u003cp\u003eWhat is the minimum number of trades that represent the total profit (possibly exceeding it).  All trade lists will represent profitable series.\u003c/p\u003e","function_template":"function y = trade_profit(x)\r\n  y = x;\r\nend","test_suite":"%%\r\ntrades = [1 3 -4 2 -1 2 3]\r\ny_correct = 2;\r\nassert(isequal(trade_profit(trades),y_correct))\r\n\r\n%%\r\ntrades = [1 2 3 -5]\r\ny_correct = 1;\r\nassert(isequal(trade_profit(trades),y_correct))\r\n\r\n%%\r\ntrades = [1 2 3 4 5 6]\r\ny_correct = 6;\r\nassert(isequal(trade_profit(trades),y_correct))\r\n\r\n%%\r\ntrades = [-2 3 -4 5 -6 1 2 3 4 5]\r\ny_correct = 3;\r\nassert(isequal(trade_profit(trades),y_correct))","published":true,"deleted":false,"likes_count":12,"comments_count":2,"created_by":240,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":617,"test_suite_updated_at":"2012-06-11T13:22:29.000Z","rescore_all_solutions":false,"group_id":6,"created_at":"2012-06-01T13:38:13.000Z","updated_at":"2026-04-05T02:56:12.000Z","published_at":"2012-06-08T19:08:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a list of results from trades made:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e[1 3 -4 2 -1 2 3]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe can add them up to see this series of trades made a profit of 6.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince the two biggest trades sum to that, we can say that all the profit came from two trades.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the minimum number of trades that represent the total profit (possibly exceeding it). All trade lists will represent profitable series.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":660,"title":"Find a subset that divides the vector into equal halves","description":"Given a vector x, return the indices to elements that will sum to exactly half of the sum of all elements.\r\n\r\nExample: \r\n\r\n Input  x  = [1 2 3 4 5 6 7]\r\n Output xi = [1 6 7]\r\n\r\nbecause \r\n\r\n sum(x) = 28\r\n sum(x([1 6 7])) = 14\r\n\r\nThe answer is not necessarily unique and the order is unimportant. We will just test to make sure that |sum(x)/2| is |sum(x(xi))|\r\n","description_html":"\u003cp\u003eGiven a vector x, return the indices to elements that will sum to exactly half of the sum of all elements.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e Input  x  = [1 2 3 4 5 6 7]\r\n Output xi = [1 6 7]\u003c/pre\u003e\u003cp\u003ebecause\u003c/p\u003e\u003cpre\u003e sum(x) = 28\r\n sum(x([1 6 7])) = 14\u003c/pre\u003e\u003cp\u003eThe answer is not necessarily unique and the order is unimportant. We will just test to make sure that \u003ctt\u003esum(x)/2\u003c/tt\u003e is \u003ctt\u003esum(x(xi))\u003c/tt\u003e\u003c/p\u003e","function_template":"function xi = split_it(x)\r\n  xi = [];\r\nend","test_suite":"%%\r\nx = [1 2 3 4 5 6 7];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%%\r\nx = [2 2 2 2 2 2];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%%\r\nx = [2     5     4     5     4];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%% \r\nx = [1     3     1     1     9     7];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%% \r\nx = primes(100);\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n","published":true,"deleted":false,"likes_count":14,"comments_count":14,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":401,"test_suite_updated_at":"2013-11-06T16:04:24.000Z","rescore_all_solutions":false,"group_id":6,"created_at":"2012-05-04T19:18:34.000Z","updated_at":"2026-02-19T13:17:58.000Z","published_at":"2012-06-08T19:08:22.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a vector x, return the indices to elements that will sum to exactly half of the sum of all elements.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  x  = [1 2 3 4 5 6 7]\\n Output xi = [1 6 7]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ebecause\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ sum(x) = 28\\n sum(x([1 6 7])) = 14]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe answer is not necessarily unique and the order is unimportant. We will just test to make sure that\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum(x)/2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum(x(xi))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":58274,"title":"Easy Sequences 116: Summation involving nthroot, floor, and ceiling functions","description":"Given a postive integers  and , we are asked to evaluate the following summation:\r\n                            \r\n                            where: the symbol \"\" and \"\" are the floor and ceil functions, respectively.\r\nFor example, if  and , we have:\r\n\u003e\u003e  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\r\n\u003e\u003e  s = S(50,3)\r\n\u003e\u003e  s =\r\n        279","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 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margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a postive integers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, we are asked to evaluate the following summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 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\" width=\"266.5\" height=\"46\" style=\"width: 266.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                            where: the symbol \"\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23.5\" height=\"19\" style=\"width: 23.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\" and \"\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23.5\" height=\"19\" style=\"width: 23.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\" are the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/matlab/ref/floor.html#\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003efloor\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/matlab/ref/ceil.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003eceil\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e functions, respectively.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFgAAAAkCAYAAAANdf2OAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAWKADAAQAAAABAAAAJAAAAADbokXeAAAEaUlEQVRoBe2Ya4iWRRTHt9J0zUhUzEsXgtVERAOpRWFTUMrUxLT8UN5FCkS0MhHFvqalfREiAovqSze60GqEWiKS9iHFD6KIuizrqlHWouUFL/X76xw5Ps6z7/usQa+7c+C3c+bMmXlmzjNzntm3qipJikCKQIpAikCKQIpAikBbInBbWzrdwn36M/cBkfkrDrUwAZ6KtMvUD56FGrgEW2AbnIEkIQI7Kf9phYU5kVqE/Txchm3wE2icczALkhCBOmgtuKdovzsSqfnYFFgx1rVPRNd42s0znP06tSOliG9Z+Xh4ExSUrGzCsCtjHEf9e7gd3oMXwcsGKvNA4z0Ke6BDyhBWrR34ccHV1+Nvu35gpO8g165gd1h5n5UrUPpIlXtq9VG7GPodo8yTEzRobH3seuY5tWe7bg76QNlOPImudDET7oI8WUqD9fkkzwn7Z87vho+kcktrUk3j0/A6rITu4OVJKm/BZG+sMH0x87nTzUm7bBJ8BE3wPMRkmDPud3pW3ecMg51eUl2Dx2lQArc3udb1Wufsf6F3c22VpL7KZH4GrcXWkS1XRSa81fnrJeXJEhpsvG/ynGJ2JXDt2PtAAdQg9iZfDjbltiOwGcqRL3HSEb1Z/HWpnOfKR7n3AZgFDWBBsXIqNi9aq7XN9g0ZfY7za/Mt4is3yBPoumArPRQVvQib9M2U44s+OOPfhfoy8Ln5AHWfMv+kbnOcgp4n02gwv+asU6esIae+Hbs9RPdCpQqVRWU1HT4v2ini/0vEVsSkwOo+rLTxTuj4MOUo2BHqZyl7BL21m4eP4R3Bv3BRSw97S43olZpvCy+MDp+6tb3gBtBxtzXPcfasqjbz25tt9Eci2+brepjSgkS7R3e+9iJfuIX0cfqvTred7EzXVN92/Jo1KOUGWMdAHzrJyKtFu/m7262kxelHne6D6MxXVN92wz8kPn9kO/q6cm7vYOhLWQOHQr1IoVvE6CIdcnynY9c16r8QO5kay+/A76jPDw8YHspY4e/LP8YcStkm4aC7sD4GlmvmluqU014ptwg/vQlhXbo1VLsG6XY9/R099qGTTVdOxeVv6A7XSakdfC/eG0A7WDeAl0CD1sEH0BnWwwr4A0pJpdwi/DxfCZUPKXVzMJG+EXRaeoF2avYj9gg2+/3ha3RLo6hxUcAngn456go7YCfYi9AD9LZ0lGR7F36A2NvF/L+L/g1uAP07q98Jst+cBdi0ngPgP3BUr4jiYDu0Pth8oTSi/i0w1Dfk6co56iB0R/wNHgSTN1CsXQldD/ft5lcppXK+zVelNsvjMALeBtkOQn/Ik9E06N4s39egS2B5sCmHj4Gy5Dm8NJDQkc/eGB7CZg9TzlKqqGR5jMk1g63Jl7IvgXLu9M/g1xjG0bqFxpJtMuRK7GjX4n0PKD2cifTUUZLPLtAOr3TpzAS1Ue4HBbMJGuAIXIByRePohdWEDocpFQP9ZpwkRSBFIEUgRSBFIEUgRSBFIEUgRcBF4F+kFEMm9jjRjQAAAABJRU5ErkJggg==\" width=\"44\" height=\"18\" style=\"width: 44px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40px; transform-origin: 404px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt;  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt;  s = S(50,3)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt;  s =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        279\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x,n)\r\n    s = sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\r\nend","test_suite":"%%\r\nx = 50; n = 3;\r\ns_correct = 279;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 100; n = 2;\r\ns_correct = 1339;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 1000:2000; n = 4;\r\ns = sum(arrayfun(@(i) S(i,n),x));\r\ns_correct = 15056572;\r\nassert(isequal(s,s_correct))\r\n%%\r\nx = 1234; n = 5;\r\ns_correct = 8508;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 10000:20000; n = 6;\r\ns = sum(arrayfun(@(i) S(i,n),x));\r\ns_correct = 1271494600;\r\nassert(isequal(s,s_correct))\r\n%%\r\nx = 123456; n = 7;\r\ns_correct = 1164369;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 100000:200000; n = 8;\r\ns = arrayfun(@(i) S(i,n),x);\r\nss = floor([mean(s),median(s),mode(s),std(s)])\r\nss_correct = [1205294 1205294 755294 259811];\r\nassert(isequal(ss,ss_correct))\r\n%%\r\nx = 1234567; n = 89;\r\ns_correct = 3703698;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 123456789; n = 10;\r\ns_correct = 1462257359;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 1234567891011; n = 12;\r\ns_correct = 23191068638973;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 11111111111111; n = 111;\r\ns_correct = 33333333333330;\r\nassert(isequal(S(x,n),s_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-05-08T15:29:34.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-05-08T10:48:38.000Z","updated_at":"2023-05-08T15:29:34.000Z","published_at":"2023-05-08T15:28:45.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a postive integers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, we are asked to evaluate the following summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                            \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es= S(x,n) = \\\\sum_{i=1}^x \\\\bigg(\\\\Big\\\\lceil \\\\sqrt[n]{i -1} \\\\Big\\\\rceil+\\\\Big\\\\lfloor \\\\sqrt[n]{i+1} \\\\Big\\\\rfloor\\\\bigg)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                            where: the symbol \\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lfloor\\\\ \\\\rfloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e\\\" and \\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lceil\\\\ \\\\rceil\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e\\\" are the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/ref/floor.html#\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/ref/ceil.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eceil\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e functions, respectively.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=50\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[\u003e\u003e  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\\n\u003e\u003e  s = S(50,3)\\n\u003e\u003e  s =\\n        279]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57512,"title":"Easy Sequences 89: Double Summation of a Trigonometric Product","description":"Given and angle  in radians and a positive integer , evaluate the following product summation:\r\n                    \r\nwhich 'directly' translates to Matlab as:\r\n    \u003e\u003e PS = @(A,x) sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));\r\nFor example  and :\r\n    \u003e\u003e x = 3; A = 1;\r\n    \u003e\u003e PS(A,x)\r\n    ans =\r\n        8.9683\r\nPlease present your answer rounded-off to nearest 4 decimal places.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 288px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 492.078125px 144px; transform-origin: 492.078125px 144px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven and angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e in radians and a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, evaluate the following product summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e                    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"236\" height=\"46\" style=\"width: 236px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhich 'directly' translates to Matlab as:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; PS = @(A,x) sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFAAAAAkCAYAAAAeor16AAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAUKADAAQAAAABAAAAJAAAAAA9sgpAAAACn0lEQVRoBe2Yz4tOURjHX5pIGgs1ERuGwljIQrJg59dqFig1iIitjcU0WUkTk5KFhYUodlKI/8HCQkxMYWoUCyULUfLr8819ppPmXvec+9535r6ep77vee45z3l+fO+959zztlouzoAz4Aw4A87ALDKwnNjHwfxZzKGdoVXHIXCinU6LfN1m8BdYW2TUgDERdxC8AKrnFqhdthHhJ1DAwdqj1RNgHm73gWdAdRhKEVjltVPgy0CtZOBP07jfvWR8GFwADzqZ/RGC2d1SW+qOdTLBhFj7g5pK1ZP6BC4m0OhfCTb1CQzL+BxelNFTCRzGuXbf00GQ9eip/gI33a+uosSv4DpYCsLXeA3XTZY9JG/11PYKXyTIdzACPoJ3wGSjKf9L2xNZ6A7sD4Cz4H02d5x2RaZrHbyf6THNVowfxUzIsX1C/66csVq6YwjU+qbPlrfgUpDNc/Sd2XXqE6g8tBxUlSVVHcTOjyHwGM43gyGgNdBEBJqkEqin+JQ5qdBOVZibNLUsgbqz54G2+VdgCzAJfdhOrNNJjHzC+FrMhLliGxZflJM2jGWZweMCw0WMrQavC2y6aqgMgfo00ffeXTCWU702jr5sTBuJExgQpQ1D590z4E3QH6ovuTACtQ7Gnim7dhfeDRn6l+UqyCOPodYE2C4FSTnS6U3oul14AUVdAd/AOVAkItBkkykR7Ti2c2EXtn+WIlLPN9XfOzrWPMw3mR7RP7h2BNIppXd6pFnKUdK1Ou6kpq67cBL8yJzdoC3abBYyfi+zteCjXDdN+kn4KbAaPqCvSyniZuDEnH2hb8MMzvRtqDGzC1sl0PGTwQw5/qtrJQY6XenbNcxfuvomQTuWF9y4OAPOgDPgDDgDzoAz4Aw4A86AM9AOBn4Dj4J2uIlkfzoAAAAASUVORK5CYII=\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 489.078125px 40px; transform-origin: 489.078125px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; x = 3; A = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; PS(A,x)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        8.9683\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ePlease present your answer rounded-off to nearest 4 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = PS(A,x)\r\n    p = sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));\r\nend","test_suite":"%%\r\nA = 1; x = 3;\r\np_correct = 8.9683;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 2:2:20; x = 3:3:30;\r\np_correct = [1.8697 -3.3674 -62.4997 11.5088 -4.8523 -71.5196 25.1022 -3.2420 -64.7730 49.3233];\r\nassert(isequal(arrayfun(@(i) PS(A(i),x(i)),1:10),p_correct))\r\n%%\r\np_correct = 4166.0746;\r\nassert(isequal(sum(arrayfun(@(i) PS(i,i),1:100)),p_correct))\r\n%%\r\nA = 100; x = 200;\r\np_correct = -739.8950;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 2000; x = 3000;\r\np_correct = 2041.1906;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 10000; x = 20000;\r\np_correct = -3131.6478;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 200000; x = 300000;\r\np_correct = -8386548.7261;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 1000000; x = 2000000;\r\np_correct = -11067350.2876;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 123456; xs = 200001:200100;\r\nps = arrayfun(@(x) PS(A,x),xs);\r\nss_correct = [-88558 -88558 -88580 12]; \r\nassert(isequal(floor([mean(ps) median(ps) mode(ps) std(ps)]),ss_correct))\r\n%%\r\nfiletext = fileread('PS.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'java') || contains(filetext, 'py') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-01-07T06:37:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-06T09:42:25.000Z","updated_at":"2025-11-22T20:00:54.000Z","published_at":"2023-01-07T06:37:48.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven and angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e in radians and a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, evaluate the following product summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e                    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ePS(A,x) = \\\\sum_{n=1}^{x} \\\\sum_{m=1}^{n} \\\\prod_{k=0}^{m-1} 2 \\\\sin \\\\left( \\\\frac_{k\\\\pi}^{m} +A \\\\right).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich 'directly' translates to Matlab as:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e PS = @(A,x) sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e x = 3; A = 1;\\n    \u003e\u003e PS(A,x)\\n    ans =\\n        8.9683]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePlease present your answer rounded-off to nearest 4 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57993,"title":"Easy Sequences 110: Integration of the Sum of a Recursive Trigonometric Function","description":"A trigonometric function, , is defined as follows:\r\n                ,  in radians\r\nApplying  recursively we define another function , for integer :\r\n                \r\nWe then define  as the sum of value of  from  to :\r\n                \r\nFinally, we are asked to evaluate the integral of  with respect to , over the real range :\r\n                \r\nFor example for , , , we have:\r\n  \u003e\u003e a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\r\n       a = 7.05797686912156\r\nPlease present the final output rounded-off to 6 decimal places. Therefore the final answer is .\r\n-------------------------\r\nNOTE: There are a number of ways to do numerical Integration in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the integral function shown above.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 507px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 462.578125px 253.5px; transform-origin: 462.578125px 253.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA trigonometric function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"31.5\" height=\"19\" style=\"width: 31.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, is defined as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"137.5\" height=\"19\" style=\"width: 137.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in radians\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eApplying \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"31.5\" height=\"19\" style=\"width: 31.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e recursively we define another function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"46\" height=\"19\" style=\"width: 46px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, for integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 51px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 25.5px; text-align: left; transform-origin: 384px 25.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-20px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"166.5\" height=\"51\" style=\"width: 166.5px; height: 51px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWe then define \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44\" height=\"19\" style=\"width: 44px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the sum of value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eR\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003e1\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQMAAABcCAYAAABqftBdAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAABA6ADAAQAAAABAAAAXAAAAAACz7CiAAAR8klEQVR4Ae2dCbQUxRWG3UEQFHEXBaIEUQjuqLigoqKI+xK3QCRR45JFIzGuMe6ixxBNIpiIawwueIgcjSzivhExeFTQKIiISkCFIKAimv/DrmO9TndPz0y/mZ6Ze8/5qeqq27eq/rp9u6rmzbDSSibGgDFgDBgDxoAxYAwYA8aAMWAMGAPGgDFgDBgDxoAxYAwYA8aAMWAMGAPGgDFgDBgDxoAxYAwYA8aAMWAMGAPGQAUYWFltrOa108LLW9YYMAbqnIH1NL4bhMeFBUI/YRvheeEr4R7BxBgwBhqAgdU1xl7CZ8IiYXthmnChsEyYL5gYA8ZAgzCwucb5tTBemCp0FdoIy4WnBBNjwBhoEAZ+pHESDL4UjgjGfEBQ9tvg2hJjwBhoAAbu1RgJBo94Y70uKOvtlVnWGDAG6piBVTS2j4TPhY7eOF9RngNF/9MFr9qyxoAxUG8M7KwBsSoY7Q1sQ69sA+UP9uosW0MMEOlNjIG0DOwfKN7v3bBbkJ+llC3EbK/OssaAMVCnDEzUuPgIcR1vfHspz2phnjDAK7esMWAMNCADnTRmOy9owIm3IRsDxoAxYAwYA8ZAvTLAl01MGouBPhruxjka8ofqy6Qc9adhu2LBoPGmnj8bztOnSIvVnw4Cf6dgUkUG7NCniuRXqeln1O4eEW1frjL+oKhUWUM3thTWFFoJmwg85JsJmwpx0loVg4Xr4xSs3BgwBpqHAR5OHno+DvQxpnmaW2F1I/17jHCT8Ibgt0t+hpCn1Yq6Y2IMNAYDh2mY4QeS67MqMHy2pv2FJwW/D4dWoG1rwhgwBiIY4C3tP4zk+Z2CnhG6zVXEny5zVkDbE5urEbNrDBgDyQywv+f3CMIBYbrK2MdXSrqpobcE+tG9Uo1aO8aAMdCUAR5ETvPDAeHWpmrNfsVhIyuEEc3ekjVgDBgDsQxwkh8OBlwfH3tH81ScKrNLhHWbx7xZNQaMgTQM/E1K4YDwX5VtkebmjHQ4WCQg8LNqJsaAMVAlBtZWu3y8Fw4IL6qMH0E1MQaMgQZiYBeNla8nhwPC0AbiwIZqDBgDAQO/UhoOBl+prJ8xZAxkzUB7GXxIuCRrwxWyN0Xt/LJCbVWjGfbt44RwQJirMv6KsJGkln21JD8t9EUlPoseIPQQ+Pvy+QKfTY8VOGA6Wmgr/EUoJO2k8JzAUnR3YaFQa9JHHX5U4A92zhHqUXjomWN+z9CXCbrgZ88IFNUQfBE/TJIvVPm+gJ+W0888+Srf9dhS6CLwB2E7CK8LQ4Q46aOKTP2UB/ZNIfyW4Ppz4WmBgMCbPo2MkRITVeunxSdqDCydvy/Uq7AtYIzhuT+vigPGb14Slkb0K9xP/JMH5kqhlE9E8uSrv9AYZgr+fKSZh8z8dGM1/rEAybOEAwW+iUZUoiMfCm4CmKBCwoC+FLi/HuQyDWKRwB/t1KtwcOjm2KWs6nap8oA3UfvzQn07V9d83+I0YZjwieD6/Jny5whpJa++eocG4Ma0Y8rBZOKnt3oN945ouLPKpgc6BIYk2VmVLN+wWS/SUgNhlcPbZ616GVRoHHykOFlwDujSmSrjo8hqyn1q3PVnWkRHWqtstKfDW3XfCL1wUZ599ZFgPLyk037DMxM/hWDI5k0Qd66wnuoIBMuFuN9F4N43BIJBJ6Ge5CwNBo4uradBhcbCEputoHvwXMpPoldT/JcVAStKCNIEbNfnZ6OUvLI8+yoPv/tCF0GuGCnLT3kjsKR3JHJoESdnqAI9viMfJXuokPqRUZU1XtZC/Z8nvCPgSPUqJ2hgzhf89MdVHDAH1q4vccGA7vmrg091nfRGzbOv7uSN93QGVoSU5ac4tn9Iw5lAm5jG26uc1UPcWcAI1TFpRwn1KKM0KMa3Tz0OzhvTyGCc7gEkXSJs7elUMps2GNynTvl9Xiehk3n21V974+iaMIa4qrL8lE8KfBKn6HqrmJY2UnlUxGW/wtKG/RpBI42sKaUBwsXCBUJ4P36AyoYKhwjlCNua7sJJwrUC0RMhEPYXrhJov62QJKeqEp7uTFKqg7rWGgPbPd8nyD9RpbGlDQazvD5zqBgnpfgqtroKZwjXC3sKvuDzQwR8qdwzlsdkA75nC0grYdsIxAW7svyUhy088bwJOLVlG5FG9pYSNggkaeQaKS0Slguu7eu8GyHclbPkg5BipYdumCxwwuxs/TswwicoE7xy6scGdXHJdwP9OXEKdVS+ncbi8wY/V1dpfGmCwXnqm5tj0lsS+lqsrxIEOFfzt9Nw41561OMTrv1LlC9V8HPH++2BkfWVPigsE1wbM5Q/ToiSsv2UiOYa8lNISLMsHhjcf1dU7yLK6DArgQ4CDztt0hbCxz2U3Sow6PFCKYL93sIYwY2JgMMkvilMFYjmjuS5yifJyqrEIQhgcYeocff3UsVHGWBcXAPNUH6ZbDreHlI+akXYDM3+n8mkYLC9tP8guH6SvipsKMTJQFWgl9ZXWSnhr4z/BsG1dazy6wlvC08KDwv4UH+hVNlfNzr7J3lGNleew92FwslC0lyU46cy/Y0MUsLb2nXGT4erHFLi5AJVoD8sTiGhnKjn2oIMIuMBCfrFVvlvDew+IzBxbFOQTwTaf56LAjJP9egyOcUIQcmNsZw0TR+L6VecbjtVuK3Cy8oTWKslfjBgxTpamCTMEnwuP9b1UKGtkCTl+GpPGXZt4uuuL2lX0En9oo7+O/ubBMq8MHlREuQISmmkoJ8WepvdplYmCL8TjhR8OUUXdOQgYalfEeQ3DdL5EXWFioiqhwVKjyrl7U2alRBgEN7MtNNaICgwjs0Et/eaqnwhwQZvAybo3ULKXv1ryp/qXZeaLabNUtvAsXFy5vt9YYDwqZAXYc62EvwAxVua1UuUb6q4iZTjq6/I0gIBn/lpkO+udJmQhfQNjExXCvcEn7EC/nO0wMs6jZTqp5G2D1Tp24KLUi7ljbpGxB0PBrpnRtQVKuoV3EsbRPtSzgfi2uDB/1zA9nsCZOLkTo5Txo3tKFeYkPJmRr9/gk6tV40MxrhYKcvwaou/MpgcdOYQpV8Jbu6eUL7Qyy64dcX+m/tK8VVsPCK4dku1gZ2wrK8CN6Yblcc3WQndIqQdm1RXSEE/TdpnOCMuZcDbCDiGLwSJY/yCIO/eHC0j6goVsQxla4C8JEBAVrKXDLngxRvhbIHzAif7BZnlSie6woTUbS04O6hHOV+DGiTglMcLU4Q8yt/VqWu8ju2p/HXedVK2HF/F7rOecVa1Wcm+MsR+H2E19lcB38Vni/W3gn4aFQx2UUNumaxsE+EBPVm4vElp9H6ety7S/pukqH+Jem6Cdi3qzsLK/tnDP6VOlPXFBQPeOJ/4FTF5Nz433hi1miwmyLu5HqL8mJyP4kL1b5LXx58pf4J3HZd1c+fmMk4vrty9uKjP0l/7eg12DPKrKv2BV54268bmxprqPvbJVxTQJIjwhvg6wMwI/dODuhERdYWK/hjc6+xvWeiGIuqnebbDE9fNq/tNSps4Av1cO6W+U2MrxD6uXIxzBjNOeSksFRjb8Ixtl2suapvgbG6gzBzB+Q5bm56uMiYtx1e3kM1Fgmvvrpg2Sil+J7A7U2kPYUlwjQ+7FYOyqaQkPx0t0wuFuNWBa3mYMo6Ap1yhl7KHo36sV5Yme7CUWKL7AeGHaW5MobOZdFyfX4jQ503i6neLqA8XtQv0cYZipbducG2Vk7IXzFo6yeBcgX4RbFYT8iRJwYB+wu0XguP1beWZqzgp1Vfh5TkB/58q0N4sIQvpIiOu/0MDg7d5ZRzcp5VUfho1yRB3uPAn4biE1tyyAxWW22GZHhTwULGSYM9ZSDaUAhPNXu9q4TSBCLiHMFJYXbhROF/4WEA6C32EKQITkiT7e5Xsv8LSNyhYqJRgsYNwnsCpbZSwL0UKtfuNVtN/X9NlHj9NYIVDAOcN+7rA2Ivdn+qWZpUWnvUoH35G9UOEGwK97yhlvvsLUX5Yiq9i+iKBh3Zb4Rzhe8LmAd5VygPbQRghIMX4qvNF7pvAP5JhwsAVuW/88uEgjy6BY2JwHU5K9lMeQBeRrlSeBzkskLtIQI+lJEuYKHlRhejwUEUJE8kEQWhL4WmBSEs5wkPG/R8IlN0sPCYQIBDsMrno4LADhSQZpUp0waYRiguCujFKOwrvCywh4+T3qsAWnNWDwPGjAmNiZdBJyKPwcNBHMCehg/d6euhek6BbyFe5Fb8/SMBXDxXwuQECcrjg+nSm8t0EtoB7C0ixvvqA7sHeZ8KagpPxyrh2jlC+nUDg6SfEScl+up8susZIXxJolL0RDxCrBRp3OkkPC3XoEaWjZLAKnR2Cyzyho6d4lVfPgwm5fv2FXj12eJPFySqq4H70olYyrYM66mcI7wkjhSTh7f65sG6SUg3VEWwZPwF+lwr3G9/6jxD34qA7awg8hMsE+ulwtvLMX1jaqGCa4PRIhwsdhbAU8lX0ZwvYWCzwErpWcMK2eong6nmxsMJ1UoyvrqqbWPlii5efL7104V6AC5V/U3jAV4jIl+ynRD86cYdwp/BhcE2Zj1d17aKespFC1CKy0WEGGBaWoM4mg981pNBZ1zxs6HwisF3wBYcliDgby5XnoY+SrVXo9C6OUlDZ857On5WP6rO7dfdAd7QrqPH0nGA8ONqxVRjL7WrzLWHlmLa7qpz5dXMYlf484l7mnU+mwvrnhnQL+SrqPFTOzt3Kh32N7YCrH6W87z+l+ur5shMW/yU5VZUEojgpy09ZKg4WiMIIk7OjwIN7mnCwQMCImzRVNRG3RBnUpPTbCyLd/kKrb4ua5DbQ1QBh/Sal315ABBPOg0zQKUdYju0tML5CMkkKBLrtCinWQP1h6qN70C6oQn8Hqk2C0EVVaNtvspCvtpcyvkhgihKeiT2F8EvN6Wbpq/g8q/gWznhMmis/pbOThbcFF2Bi+l1ycUfdyQoiKoqWbDThxn1V97VwRoJOrVSxLF8sMJ7bhErL6WqQQEAw2rzSjYfaqzdfzaWfdhLpHwl3hsjP4rKHjMwWxgn+QUsWtqNsbKLCd4X7oyprrKyD+jtHIBA8LjRXsJbpJsLy+nDhWYG2wXghD9JJnagHX821nx4kknkDXCpkKTjz9cKqWRqNsbWWyqcIrHLWjtGplWLG8i+BB/FNoRKHoC3VzinCG4ILAi49QWV5kVr31Zrw00GabZbznD9kJZ2yMpTCDp+/8+BsmUI3zyoETsbCg8hbsIuQhawuIxzEseLYSuDwiod/mDBRcCfkLgC4dIHqKrGqUzOpZZA0a9VXa8ZPu4vkWj10O1J9byvUuvBwugeRlDODcvFFyKZvv1B+uO7No9Sqr9aLn+bRJ+qqTxzYFXo4K13Px24mxoAxUGEGOO+o9MOe1B5/FGRiDBgDxoAxYAwYA8aAMWAMGAPGgDFgDBgDxoAxYAwYA8aAMWAMGAPGgDGQewbSfvMw9wOxDpbFwE66u7+wjcDftPcWshb+OvFkgT9J5o+eTHLGAF8YMTEGPhAF/DDHUcLcjOlwX4nnz7hvFrplbN/MZcSABYOMiKxxM/yqE9/8RPgOQZbCt0o/Ff6RpVGzlT0DFgyy57RWLe4VdHxCxgN4WfZGCU9nbNfMZcyABYOMCa1hc33Vd74OzteLTRqQAQsGDTjpEUPurLItBLdFuFt5fp3qReEKwaQBGOBwx8QYYFWAuC3CU8rvI5woECBGChwuppXBUrw3rbLp5YMBCwb5mIdq98IFAx58vuLMrw5tK7hPFp5Unt8rSCvvpFU0PWPAGMgPA2wV5wtvCTcJ/BZhKyFrIcDwVWY+XjTJIQN2ZpDDSalwl1gB8BPgnBnwa8/LBP7mwKTBGLBg0GATHjFct0X4iepeEPoJXUJ6nBksKgL8r0cmNcaAnRnU2IQ1Q3cJBvxqNQd+rAj4T23478oIDj2FGUKxZwb8RaOJMWAM1BADfE9gqcDHiMjqwkyBw8JjhUlCVucH7kdYH5ZNE2PAGMgZA3uqPxzqXeX1i23Cl8JCYR+vvNQsWw4+pSDA0NZy4R4hC9syY2IMGAPNyUAHGW/TnA2YbWPAGDAGjAFjwBgwBowBY8AYMAaMAWPAGDAGjAFjwBgwBowBY8AYMAaMgVpj4H+fNy4XICogbQAAAABJRU5ErkJggg==\" width=\"129.5\" height=\"46\" style=\"width: 129.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFinally, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ewe are asked to evaluate the integral of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e with respect to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, over the real range \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-21px\"\u003e\u003cimg 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width=\"204\" height=\"48\" style=\"width: 204px; height: 48px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42.5\" height=\"20\" style=\"width: 42.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGQAAAAoCAYAAAAIeF9DAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAZKADAAQAAAABAAAAKAAAAACyuGsJAAAEwklEQVRoBe2Ya6gVVRSATb1a9MeyrCj6cUMtLIv+lEWBJgY96UcKIWmE9aMQI4x+GFekMnobUVHRNcgoAskgi8qKHhglkoJJaS8rMETTQulhj++r2TBMs8+Z25lzOkf3gu/uNWuvWbNm7cfsc4cNS5IqkCqQKpAqkCqQKpAqkCpQfwUOqT9kT0fsI/upMBkmwJewFj6EfZCkgxU4nWd9DH+WsBPbWZCkQxUYz3NcAWWDEWw/0T+tQ/kc1I9x234HfoelcByMgJOy619pw6CsQ0/S5gpMIb4FXxR5zrysXx8H7YiIXzLXVIE7iLMZhkfiuYK2QFgll0X8ajHHkqgleI8EGUOei+GPSL4OxIZcX8wv5/Lf1ZENbp1I33Toh1XgPhtkLMq14PK9C/ZAr8oNFRL/OefzRU4vU53kx8LR0OhnxXb6pak4EC7h/RCWqQk5CIr930HoG9B4gMt7vJ/v+2mD95xE3zPwC4TaNGr1rSSH4+WPIkf6AQhBZ6EfBZ+Dq2U1fA8XQ13iWd8zf6u8VldCxBkHv4F1WABlMgOjx+JQqyrtnLJAzWz+WArBl6GvhLegD9oh5xI0PK+V9oMak1uY5bSBtuy9nbx7wQm6GELe96NfA/Mzm8dnt3lt4nb/L2m0x+ls/y7ww6fshlPBLasd4nNm1hB4GzFerSGOv0ncvkeDk2U9FMXt7Fl4BE6BT8Di+w35EebCIKyB6dCyvEKEMOo3thytdwI4GVeCp6orG6Q9Ktd3H7q1ejlneyOz3ZSztaTelgX0QZMbRHLp3gMu3SdhNjRbgbh0rSwhM995YcUMHZgd2T3zsnuOp/XHpHGsTy1iQgaU6yMRL8S+L/MJvrZPgYeDXpOrSdj8lw4h8ZnZPa4otzrlFjDOVi/qEP+fkz89lB3VDsPnW3B1OAvOhAchDMx56FWlG05Zl5Osp6pHqyad+Xmy8539V32QjSjaHg6GVtqR3LwW3gVPGAb+GopyCYbni0aunwPvGSjpi5n+71PWBSTmb64VEFvZno5uh7ycxoUrw/e9NeuYlF1ruzSzjaN1G4uKRY+J347xcAbcDH4/TszwFHMRnAAuR32L8hGGWeAKqyqbcIxti1Vj6Gd+Q5WzuWEVONPngAUuSh+G5eD2HOQYlMfB76UraxCUGf80f//9jL/2PwEvQvBBjUs/XRb5UHDZ7ocwslegO9LiScvj3U6YCjExSf0d0G4XJ9suMF9X+/ISXsLmO+szDZS58ANoE3eFII+hBPvb6G/CV+CgVpJv8DLAXnB23A1BxqA4K0L/bvR7Q2ek3YzdPdSZ0c3id287hOI1a90RfKfRsCd3n3VzmwpyJ0ox1pBWv9tFCLACfXiInLVhxuvjLBpR6M9fOoPcqk7OG7tUf528wntXacM3wp1kR+7eqwrvdw7X4chru6TQ3/RyLB5uURMjns6K82FKpD+Y/ehtgdnBcAC3bnUDkP9e5F/Xrd3PwJF5Yyf1UTzMvXLIs6GTSR4sz3ILewEeKnlhl3eSDlbA7WwQngb1vFzHxaK8IenxChSLF/ds3LOM7vnwPnhcDuIPoQnQD9uCMbXtrYBHuUank9XtfXyKniqQKpAqkCqQKpAqkCqQKpAqkCrQYxX4CzYWfmZMLQJmAAAAAElFTkSuQmCC\" width=\"50\" height=\"20\" style=\"width: 50px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 459.578125px 20px; transform-origin: 459.578125px 20px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 459.578125px 10px; transform-origin: 459.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 459.578125px 10px; transform-origin: 459.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e       a = 7.05797686912156\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"85.5\" height=\"18\" style=\"width: 85.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e-------------------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThere are a number of ways to do \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/matlab/numerical-integration-and-differentiation.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumerical Integration\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e function shown above.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function a = A(n,x1,x2) \r\n    y = x;\r\nend","test_suite":"%%\r\n[n,x1,x2] = deal(1:10,pi,2*pi);\r\na_correct = [3.065357 5.25927 7.057977 8.618935 10.016842 11.294017 12.477158 13.584381 14.628646 15.619599];\r\nassert(all(abs(arrayfun(@(i) A(i,x1,x2),n)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(12,exp(1),exp(1.5));\r\na_correct = 9.752678;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(15,-2,10);\r\na_correct = 50.909769;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(25,-10*pi,0);\r\na_correct = -267.631308;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(1000,0,1000);\r\na_correct = 61793.524569;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(10000,5,1234);\r\na_correct = 244011.112390;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(123456,10000,12345);\r\na_correct = 1644471.557504;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(1:1000,-pi,20*exp(1));\r\na = arrayfun(@(i) A(i,x1,x2),n);\r\ns = round([sum(a) sum(diff(a,1)) sum(diff(a,2)) sum(diff(a,3))]);\r\ns_correct = [2087798 3114 -35 7];\r\nassert(isequal(s,s_correct))\r\n%%\r\n[n,x1,x2] = deal(randi(15),rand(),100*rand());\r\nt = 'sin(atan(';\r\na_correct = 0;\r\nfor i = 1:n\r\n    a_correct = a_correct + integral(str2num(['@(x)' repmat('sin(atan(',1,i) 'x' repmat('))',1,i)]),x1,x2);\r\nend\r\na_correct = round(a_correct,6);\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\nfiletext = fileread('A.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'assignin') || contains(filetext, 'evalin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-04-23T09:34:40.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-04-16T08:49:13.000Z","updated_at":"2023-04-23T09:34:40.000Z","published_at":"2023-04-16T17:57:39.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA trigonometric function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is defined as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x) = \\\\sin(\\\\arctan(x))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in radians\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eApplying \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e recursively we define another function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{R}(x,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{R}(x,n)=\\\\underbrace{\\\\text{T}(\\\\text{T}(\\\\text{T}(...\\\\text{T}(x))))}\\\\\\\\_{\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\text{n times}}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe then define \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{S}(x,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the sum of value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{R}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{S}(x,n) = \\\\sum_{k=1}^{n} \\\\text{R}(x,k) \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFinally, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ewe are asked to evaluate the integral of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{S}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e with respect to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, over the real range \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[x_1,x_2]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea=\\\\text{A}(n,x_1,x_2) = \\\\int^{x_2}_{x_1} \\\\text{S}(x,n)\\\\ \\\\mathrm{d}x}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1=\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2=2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\\n       a = 7.05797686912156]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea=7.057977\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-------------------------\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eThere are a number of ways to do \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/numerical-integration-and-differentiation.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumerical Integration\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e function shown above.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57983,"title":"Easy Sequences 109: Summation of Derivatives of a Trigonometric Function","description":"A trigonometric function, ,  is defined as follows:\r\n                \r\n                where:     ; and \r\n                                 is in radians.\r\nIn this problem we are asked to evaluate the following summation:\r\n                \r\n                where:     is a real number; \r\n                                 is an integer; \r\n                                 or the -th derivative of  with respect to ; and \r\n                                .\r\nFor example for  and , we have:\r\n  \u003e\u003e  syms x;\r\n  \u003e\u003e  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\r\n  \u003e\u003e  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\r\n  \u003e\u003e  s = vpa(subs(S,x,2))\r\n        s = 0.10315887444431633673347091141408;\r\nPlease present the final output rounded-off to 6 decimal places. Therefore the final answer is .\r\n-------------------------\r\nNOTE: Symbolic toolbox is not available to Cody players. It is possible to do numerical differentiation in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using syms, as shown above.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 622px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 311px; transform-origin: 407px 311px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA trigonometric function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"31.5\" height=\"19\" style=\"width: 31.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,  is defined as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19px; text-align: left; transform-origin: 384px 19px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-14px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"133.5\" height=\"38\" style=\"width: 133.5px; height: 38px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                where:     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"152.5\" height=\"20\" style=\"width: 152.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is in radians.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eIn this problem we are asked to evaluate the following summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"142.5\" height=\"46\" style=\"width: 142.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                where:    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is a real number; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is an integer; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"112.5\" height=\"37\" style=\"width: 112.5px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e-th derivative of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with respect to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"91\" height=\"20\" style=\"width: 91px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 100px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 50px; transform-origin: 404px 50px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  syms x;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  s = vpa(subs(S,x,2))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        s = 0.10315887444431633673347091141408;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"83.5\" height=\"18\" style=\"width: 83.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e-------------------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSymbolic toolbox is not available to Cody players. It is possible to do \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.unioviedo.es/compnum/labs/lab07_der_int/lab07_der_int.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumerical differentiation\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003esyms\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, as shown above.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x,n)\r\n  s = x;\r\nend","test_suite":"%%\r\nx = 1:10; n = 2;\r\ns_correct = [-1.325444 0.103159 1.239586 -1.134858 -0.29505 1.380427 -0.85387 -0.669756 1.411304 -0.504863];\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 11:20; n = 2*x;\r\ns_correct = [-0.991110 -0.481399 -0.115639 -0.691700 1.142283 1.385650 -1.377653 -1.119743 0.658705  0.078175];\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 25.25; n = 25;\r\ns_correct = 1.945253;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 100; n = 100;\r\ns_correct = -0.386110;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 0.000123; n = 123;\r\ns_correct = 0.000001;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 1234; n = 1:1000;\r\na_correct = [275 250];\r\nassert(isequal([round(sum(S(x,n))) sum(round(S(x,n)))],a_correct))\r\n%%\r\nx = 123456; n = 123456;\r\ns_correct = 0.899338;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 123456789.10111213; n = 123456789;\r\ns_correct = -1.993727;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = rand()*randi(1000); n = randi(10)-1;\r\nTs = { \r\n       @(a) (2*tan(a) - tan(a)^2 + 1)/(tan(a)^2 + 1);\r\n       @(a) (2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2)/(tan(a)^2 + 1) - (2*tan(a)*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1);\r\n       @(a) 2*tan(a)^2 - 4*tan(a) - (4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2)/(tan(a)^2 + 1) + (4*tan(a)^2*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (4*tan(a)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - 2;\r\n       @(a) 8*tan(a)*(2*tan(a) - tan(a)^2 + 1) - 12*tan(a)^2 - (16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2)/(tan(a)^2 + 1) + 12*tan(a)*(tan(a)^2 + 1) + (12*tan(a)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (8*tan(a)^3*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (6*tan(a)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - 12;\r\n       @(a) 8*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) - (16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3)/(tan(a)^2 + 1) + 48*tan(a)^2*(tan(a)^2 + 1) - 24*tan(a)^2*(2*tan(a) - tan(a)^2 + 1) - 48*tan(a)*(tan(a)^2 + 1) + 32*tan(a)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 24*(tan(a)^2 + 1)^2 - (24*tan(a)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (32*tan(a)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + (8*tan(a)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (16*tan(a)^4*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1);\r\n       @(a) 320*tan(a)*(tan(a)^2 + 1)^2 - 160*tan(a)^2*(tan(a)^2 + 1) + 160*tan(a)^3*(tan(a)^2 + 1) - 120*tan(a)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - (272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3)/(tan(a)^2 + 1) + 40*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 64*tan(a)^3*(2*tan(a) - tan(a)^2 + 1) - 80*tan(a)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 80*(tan(a)^2 + 1)^2 + (80*tan(a)^3*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (80*tan(a)^4*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (32*tan(a)^5*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (40*tan(a)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (10*tan(a)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1);\r\n       @(a) 480*tan(a)^4*(tan(a)^2 + 1) - 960*tan(a)*(tan(a)^2 + 1)^2 - 480*tan(a)^3*(tan(a)^2 + 1) - (64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4)/(tan(a)^2 + 1) + 360*tan(a)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 384*tan(a)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 160*tan(a)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 32*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) + 2640*tan(a)^2*(tan(a)^2 + 1)^2 - 120*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 160*tan(a)^4*(2*tan(a) - tan(a)^2 + 1) + 480*(tan(a)^2 + 1)^3 - (240*tan(a)^4*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (60*tan(a)^2*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - (192*tan(a)^5*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + 128*tan(a)^2*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) + (12*tan(a)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 192*tan(a)*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (64*tan(a)^6*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (160*tan(a)^3*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1);\r\n       @(a) 11424*tan(a)*(tan(a)^2 + 1)^3 - 280*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 224*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 1344*tan(a)^4*(tan(a)^2 + 1) + 1344*tan(a)^5*(tan(a)^2 + 1) - 1344*tan(a)^3*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 1120*tan(a)^4*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 7392*tan(a)^2*(tan(a)^2 + 1)^2 + 17472*tan(a)^3*(tan(a)^2 + 1)^2 - (7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4)/(tan(a)^2 + 1) + 384*tan(a)^5*(2*tan(a) - tan(a)^2 + 1) + 840*tan(a)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 280*tan(a)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 1344*(tan(a)^2 + 1)^3 + (672*tan(a)^5*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (280*tan(a)^3*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) + (448*tan(a)^6*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + 128*tan(a)*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) - 384*tan(a)^3*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) + 672*tan(a)*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (128*tan(a)^7*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (560*tan(a)^4*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (84*tan(a)^2*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) + 896*tan(a)^2*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (14*tan(a)*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1);\r\n       @(a) 3584*tan(a)^6*(tan(a)^2 + 1) - 30464*tan(a)*(tan(a)^2 + 1)^3 - 3584*tan(a)^5*(tan(a)^2 + 1) - (256*tan(a)^8*(tan(a)^2 + 1) - 256*tan(a)^7*(tan(a)^2 + 1) - 15872*tan(a)*(tan(a)^2 + 1)^4 + 137216*tan(a)^2*(tan(a)^2 + 1)^4 - 49152*tan(a)^3*(tan(a)^2 + 1)^3 + 185856*tan(a)^4*(tan(a)^2 + 1)^3 - 15360*tan(a)^5*(tan(a)^2 + 1)^2 + 31616*tan(a)^6*(tan(a)^2 + 1)^2 + 7936*(tan(a)^2 + 1)^5)/(tan(a)^2 + 1) + 4480*tan(a)^4*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 1680*tan(a)^2*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) + 3072*tan(a)^5*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 448*tan(a)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) + 128*(tan(a)^2 + 1)^3*(2*tan(a) - tan(a)^2 + 1) + 161280*tan(a)^2*(tan(a)^2 + 1)^3 - 46592*tan(a)^3*(tan(a)^2 + 1)^2 + 102144*tan(a)^4*(tan(a)^2 + 1)^2 - 560*(tan(a)^2 + 1)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 896*tan(a)^6*(2*tan(a) - tan(a)^2 + 1) - 3584*tan(a)^3*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 15232*(tan(a)^2 + 1)^4 + 896*(tan(a)^2 + 1)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (1792*tan(a)^6*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (1120*tan(a)^4*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - (1024*tan(a)^7*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (112*tan(a)^2*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) + 1152*tan(a)^4*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) + (16*tan(a)*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) - 640*tan(a)^2*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) + (256*tan(a)^8*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (1792*tan(a)^5*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (448*tan(a)^3*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 3584*tan(a)^2*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 1024*tan(a)*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 3072*tan(a)^3*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 1792*tan(a)*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2);\r\n       @(a) 1152*(tan(a)^2 + 1)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 571392*tan(a)*(tan(a)^2 + 1)^4 - 9216*tan(a)^6*(tan(a)^2 + 1) + 9216*tan(a)^7*(tan(a)^2 + 1) - 1008*(tan(a)^2 + 1)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) - 13824*tan(a)^5*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 8064*tan(a)^3*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 8064*tan(a)^6*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 2688*(tan(a)^2 + 1)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 414720*tan(a)^2*(tan(a)^2 + 1)^3 + 1769472*tan(a)^3*(tan(a)^2 + 1)^3 - 262656*tan(a)^4*(tan(a)^2 + 1)^2 + 552960*tan(a)^5*(tan(a)^2 + 1)^2 - (353792*tan(a)*(tan(a)^2 + 1)^5 - 512*tan(a)^8*(tan(a)^2 + 1) + 512*tan(a)^9*(tan(a)^2 + 1) - 274432*tan(a)^2*(tan(a)^2 + 1)^4 + 1841152*tan(a)^3*(tan(a)^2 + 1)^4 - 371712*tan(a)^4*(tan(a)^2 + 1)^3 + 1304832*tan(a)^5*(tan(a)^2 + 1)^3 - 63232*tan(a)^6*(tan(a)^2 + 1)^2 + 128512*tan(a)^7*(tan(a)^2 + 1)^2 - 15872*(tan(a)^2 + 1)^5)/(tan(a)^2 + 1) + 2048*tan(a)^7*(2*tan(a) - tan(a)^2 + 1) + 13440*tan(a)^4*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 3024*tan(a)^2*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) - 39168*(tan(a)^2 + 1)^4 - 672*tan(a)*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4) + (4608*tan(a)^7*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (4032*tan(a)^5*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (2304*tan(a)^8*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + (672*tan(a)^3*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) - 512*tan(a)*(tan(a)^2 + 1)^3*(2*tan(a) - tan(a)^2 + 1) - 3072*tan(a)^5*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) - 10752*tan(a)^2*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 4032*tan(a)*(tan(a)^2 + 1)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) + 2048*tan(a)^3*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) - (512*tan(a)^9*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (5376*tan(a)^6*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (2016*tan(a)^4*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 4608*tan(a)*(tan(a)^2 + 1)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 13824*tan(a)^3*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (144*tan(a)^2*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) + 10368*tan(a)^4*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (18*tan(a)*(256*tan(a)^8*(tan(a)^2 + 1) - 256*tan(a)^7*(tan(a)^2 + 1) - 15872*tan(a)*(tan(a)^2 + 1)^4 + 137216*tan(a)^2*(tan(a)^2 + 1)^4 - 49152*tan(a)^3*(tan(a)^2 + 1)^3 + 185856*tan(a)^4*(tan(a)^2 + 1)^3 - 15360*tan(a)^5*(tan(a)^2 + 1)^2 + 31616*tan(a)^6*(tan(a)^2 + 1)^2 + 7936*(tan(a)^2 + 1)^5))/(tan(a)^2 + 1);\r\n       @(a) 6720*(tan(a)^2 + 1)^2*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 1428480*tan(a)*(tan(a)^2 + 1)^4 - 23040*tan(a)^7*(tan(a)^2 + 1) + 23040*tan(a)^8*(tan(a)^2 + 1) + 40320*tan(a)^6*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (1024*tan(a)^10*(tan(a)^2 + 1) - 1024*tan(a)^9*(tan(a)^2 + 1) - 707584*tan(a)*(tan(a)^2 + 1)^5 + 9061376*tan(a)^2*(tan(a)^2 + 1)^5 - 3682304*tan(a)^3*(tan(a)^2 + 1)^4 + 21253376*tan(a)^4*(tan(a)^2 + 1)^4 - 2609664*tan(a)^5*(tan(a)^2 + 1)^3 + 8728576*tan(a)^6*(tan(a)^2 + 1)^3 - 257024*tan(a)^7*(tan(a)^2 + 1)^2 + 518656*tan(a)^8*(tan(a)^2 + 1)^2 + 353792*(tan(a)^2 + 1)^6)/(tan(a)^2 + 1) + 33600*tan(a)^4*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) + 20480*tan(a)^7*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 5040*tan(a)^2*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4) - 512*(tan(a)^2 + 1)^4*(2*tan(a) - tan(a)^2 + 1) - 960*tan(a)*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4) + 12349440*tan(a)^2*(tan(a)^2 + 1)^4 - 4423680*tan(a)^3*(tan(a)^2 + 1)^3 + 16727040*tan(a)^4*(tan(a)^2 + 1)^3 - 1382400*tan(a)^5*(tan(a)^2 + 1)^2 + 2845440*tan(a)^6*(tan(a)^2 + 1)^2 - 1680*(tan(a)^2 + 1)*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4) - 4608*tan(a)^8*(2*tan(a) - tan(a)^2 + 1) - 46080*tan(a)^5*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 16128*tan(a)^3*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) + 714240*(tan(a)^2 + 1)^5 - 5760*(tan(a)^2 + 1)^3*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 8064*tan(a)*(tan(a)^2 + 1)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) + 28800*tan(a)^2*(tan(a)^2 + 1)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (11520*tan(a)^8*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (13440*tan(a)^6*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) + 20480*tan(a)^3*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - (5120*tan(a)^9*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (3360*tan(a)^4*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) + 8192*tan(a)^6*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) - 15360*tan(a)*(tan(a)^2 + 1)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 46080*tan(a)^3*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - (180*tan(a)^2*(256*tan(a)^8*(tan(a)^2 + 1) - 256*tan(a)^7*(tan(a)^2 + 1) - 15872*tan(a)*(tan(a)^2 + 1)^4 + 137216*tan(a)^2*(tan(a)^2 + 1)^4 - 49152*tan(a)^3*(tan(a)^2 + 1)^3 + 185856*tan(a)^4*(tan(a)^2 + 1)^3 - 15360*tan(a)^5*(tan(a)^2 + 1)^2 + 31616*tan(a)^6*(tan(a)^2 + 1)^2 + 7936*(tan(a)^2 + 1)^5))/(tan(a)^2 + 1) + (20*tan(a)*(353792*tan(a)*(tan(a)^2 + 1)^5 - 512*tan(a)^8*(tan(a)^2 + 1) + 512*tan(a)^9*(tan(a)^2 + 1) - 274432*tan(a)^2*(tan(a)^2 + 1)^4 + 1841152*tan(a)^3*(tan(a)^2 + 1)^4 - 371712*tan(a)^4*(tan(a)^2 + 1)^3 + 1304832*tan(a)^5*(tan(a)^2 + 1)^3 - 63232*tan(a)^6*(tan(a)^2 + 1)^2 + 128512*tan(a)^7*(tan(a)^2 + 1)^2 - 15872*(tan(a)^2 + 1)^5))/(tan(a)^2 + 1) + 3072*tan(a)^2*(tan(a)^2 + 1)^3*(2*tan(a) - tan(a)^2 + 1) - 7168*tan(a)^4*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) + (1024*tan(a)^10*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (15360*tan(a)^7*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (8064*tan(a)^5*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 51840*tan(a)^4*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + (960*tan(a)^3*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) - 26880*tan(a)^2*(tan(a)^2 + 1)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 5120*tan(a)*(tan(a)^2 + 1)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 30720*tan(a)^5*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2);\r\n     };     \r\ns = S(x,n);\r\ns_correct = 0;\r\nfor  i = 1:n+1\r\n    s_correct = s_correct + Ts{i}(x) / 2^(i-1);\r\nend\r\ns_correct = round(s_correct,6);\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nfiletext = fileread('S.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'assignin') || contains(filetext, 'evalin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-04-16T12:26:21.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2023-04-15T15:36:04.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2023-04-14T18:29:33.000Z","updated_at":"2023-04-16T12:26:21.000Z","published_at":"2023-04-15T14:27:33.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA trigonometric function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,  is defined as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)=\\\\frac^{1+2\\\\tan(x)-\\\\tan^2(x)}_{1+\\\\tan^2(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                where:     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tan^2(x) = \\\\tan(x)\\\\cdot\\\\tan(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is in radians.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIn this problem we are asked to evaluate the following summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es=\\\\text{S}(x,n)=\\\\sum_{k=0}^n \\\\frac^{\\\\text{T}^{(k)}(x)}_{2^k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                where:    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a real number; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is an integer; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}^{(k)}(x)=\\\\frac{\\\\mathrm{d}^k}{\\\\mathrm{d}x^k}\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e-th derivative of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with respect to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}^{(0)}(x)=\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e  syms x;\\n  \u003e\u003e  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\\n  \u003e\u003e  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\\n  \u003e\u003e  s = vpa(subs(S,x,2))\\n        s = 0.10315887444431633673347091141408;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es=0.103159\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-------------------------\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eSymbolic toolbox is not available to Cody players. It is possible to do \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.unioviedo.es/compnum/labs/lab07_der_int/lab07_der_int.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumerical differentiation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esyms\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, as shown above.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8071,"title":"summation of the reciprocals ","description":"Determine the summation of the reciprocals of numbers with length equal to N,in the meanwhile there is a digit K excluded from the numbers. \r\n\r\nreference:( \u003chttp://en.wikipedia.org/wiki/Kempner_series\u003e )\r\n\r\nFor example:\r\n\r\nif N = 2 and K = 7 \r\n\r\nthe matrix are as following:\r\n\r\n    10    20    30    40    50    60    80    90\r\n    11    21    31    41    51    61    81    91\r\n    12    22    32    42    52    62    82    92\r\n    13    23    33    43    53    63    83    93\r\n    14    24    34    44    54    64    84    94\r\n    15    25    35    45    55    65    85    95\r\n    16    26    36    46    56    66    86    96\r\n    18    28    38    48    58    68    88    98\r\n    19    29    39    49    59    69    89    99\r\n\r\nouput then can be the summation of the reciprocals of above matrix\r\n\r\noutput = 2.01554407485017\r\n\r\nSo, give the input N and K, determine the output as mentioned above  \r\n\r\n\r\n\r\n\r\n\r\n ","description_html":"\u003cp\u003eDetermine the summation of the reciprocals of numbers with length equal to N,in the meanwhile there is a digit K excluded from the numbers.\u003c/p\u003e\u003cp\u003ereference:( \u003ca href = \"http://en.wikipedia.org/wiki/Kempner_series\"\u003ehttp://en.wikipedia.org/wiki/Kempner_series\u003c/a\u003e )\u003c/p\u003e\u003cp\u003eFor example:\u003c/p\u003e\u003cp\u003eif N = 2 and K = 7\u003c/p\u003e\u003cp\u003ethe matrix are as following:\u003c/p\u003e\u003cpre\u003e    10    20    30    40    50    60    80    90\r\n    11    21    31    41    51    61    81    91\r\n    12    22    32    42    52    62    82    92\r\n    13    23    33    43    53    63    83    93\r\n    14    24    34    44    54    64    84    94\r\n    15    25    35    45    55    65    85    95\r\n    16    26    36    46    56    66    86    96\r\n    18    28    38    48    58    68    88    98\r\n    19    29    39    49    59    69    89    99\u003c/pre\u003e\u003cp\u003eouput then can be the summation of the reciprocals of above matrix\u003c/p\u003e\u003cp\u003eoutput = 2.01554407485017\u003c/p\u003e\u003cp\u003eSo, give the input N and K, determine the output as mentioned above\u003c/p\u003e","function_template":"function output = sum_reciprocals(N,K)\r\n  output  = [];\r\nend","test_suite":"%%\r\nN = 2;\r\nK = 7;\r\nassert(abs(sum_reciprocals(N,K)-2.01554407485016762)\u003c1e-8)\r\n%%\r\nN = 5;\r\nK = 7;\r\nassert(abs(sum_reciprocals(N,K)-1.440856832360981609753)\u003c1e-8)\r\n%%\r\nN = 8;\r\nK = 9;\r\nassert(abs(sum_reciprocals(N,K)-1.0714523172876426748)\u003c1e-8)\r\n%%\r\nN = 2:8;\r\nK = 9;\r\nM = arrayfun(@(x)sum_reciprocals(x,K),N);\r\nR = spline(N,M,[3.3 5.93]);\r\nassert(all(abs(R-[1.7588093884770861, 1.3325666476222477463])\u003c1e-8))","published":true,"deleted":false,"likes_count":3,"comments_count":1,"created_by":13709,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":"2015-04-08T14:05:58.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2015-04-08T11:21:46.000Z","updated_at":"2015-04-10T00:50:25.000Z","published_at":"2015-04-08T11:21:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the summation of the reciprocals of numbers with length equal to N,in the meanwhile there is a digit K excluded from the numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ereference:(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Kempner_series\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Kempner_series\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eif N = 2 and K = 7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe matrix are as following:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    10    20    30    40    50    60    80    90\\n    11    21    31    41    51    61    81    91\\n    12    22    32    42    52    62    82    92\\n    13    23    33    43    53    63    83    93\\n    14    24    34    44    54    64    84    94\\n    15    25    35    45    55    65    85    95\\n    16    26    36    46    56    66    86    96\\n    18    28    38    48    58    68    88    98\\n    19    29    39    49    59    69    89    99]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eouput then can be the summation of the reciprocals of above matrix\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eoutput = 2.01554407485017\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSo, give the input N and K, determine the output as mentioned above\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":57517,"title":"Easy Sequences 90: Triple Summation of a Combinatorial Function","description":"Given a positive integer , we are asked to evaluate the following triple summation:\r\n                \r\nThe symbol  is the combination function, nchoosek in Matlab. Therefore, we can write the function in Matlab as follows:\r\n    \u003e\u003e S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));\r\nHence, for , we have:\r\n    \u003e\u003e S(5)\r\n    \u003e\u003e ans =\r\n       -1.3302\r\nPlease round-off the output to the nearest 4 decimal places.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 313px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 563.71875px 156.5px; transform-origin: 563.71875px 156.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, we are asked to evaluate the following triple summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 57px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 28.5px; text-align: left; transform-origin: 384px 28.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; 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\" width=\"169\" height=\"57\" style=\"width: 169px; height: 57px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 55px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 27.5px; text-align: left; transform-origin: 384px 27.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-14px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"34\" style=\"width: 28.5px; height: 34px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the combination function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003enchoosek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in Matlab. Therefore, we can write the function in Matlab as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eHence, for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 560.71875px 30px; transform-origin: 560.71875px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; S(5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e       -1.3302\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ePlease round-off the output to the nearest 4 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x)\r\n    s = sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));\r\nend","test_suite":"%%\r\nx = 1:20;\r\ns_correct = [1.0000 0.6667 0.1333 -0.5429 -1.3302 -2.2084 -3.1635 -4.1853 -5.2659 -6.3992 -7.5801 -8.8044 -10.0688 -11.3702 -12.706 -14.0742 -15.4726 -16.8996 -18.3536 -19.8333];\r\nassert(isequal(arrayfun(@S,x),s_correct))\r\n%%\r\nx = 25;\r\ns_correct = -27.5763;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 50;\r\ns_correct = -72.3576;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 100;\r\ns_correct = -179.0764;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1000;\r\ns_correct = -2936.8505;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 10000;\r\ns_correct = -40871.0455;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 100000;\r\ns_correct = -523824.1437;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1000000;\r\ns_correct = -6389513.2367;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nxs = 1000001:2000000;\r\ns = arrayfun(@(x) S(x),xs);\r\nss_correct = [-9902420 -9888371 -13472171 2045750];\r\nassert(isequal(floor([mean(s) median(s) mode(s) std(s)]),ss_correct))\r\n%%\r\nfiletext = fileread('S.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'java') || contains(filetext, 'py') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-01-07T10:58:56.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-07T09:14:32.000Z","updated_at":"2023-01-07T10:58:57.000Z","published_at":"2023-01-07T10:58:56.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, we are asked to evaluate the following triple summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(x) = \\\\sum_{y=0}^{x} \\\\sum_{n=0}^{y} \\\\sum_{k=0}^n \\\\frac_{(-4)^{k} {n \\\\choose k}}^{2k \\\\choose k}.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en \\\\choose k\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the combination function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enchoosek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e in Matlab. Therefore, we can write the function in Matlab as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHence, for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e S(5)\\n    \u003e\u003e ans =\\n       -1.3302]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePlease round-off the output to the nearest 4 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57507,"title":"Easy Sequences 88: Double Summation of a Combinatorial Function","description":"Given two positive integers  and , , we define the function  as the sum of the reciprocals of the products of all combinations of  to  taken  at a time. In Matlab we say that:    R = sum(1./prod(nchoosek(a,b)')). For example if  and :\r\n  \u003e\u003e R = sum(1./prod(nchoosek(1:5,3)'))\r\n  R =\r\n      0.7083\r\nGiven a positive integer , we are asked to evaluate the following double summation:\r\n                \r\nFor example for :\r\n                \r\nIn which, in Matlab, evaluates to:\r\n  \u003e\u003e S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\r\n  S =\r\n      5.5833\r\nPlease round-off  to the nearest whole number. Therefore, the final output for the above example is . ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 435px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 757.578125px 217.5px; transform-origin: 757.578125px 217.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eGiven two positive integers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we define the function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"46\" height=\"19\" style=\"width: 46px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the sum of the reciprocals of the products of all combinations of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003e1\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e taken \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e at a time. In Matlab we say that:    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eR = sum(1./prod(nchoosek(a,b)'))\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. For example if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 754.578125px 30px; transform-origin: 754.578125px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; R = sum(1./prod(nchoosek(1:5,3)'))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  R =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e      0.7083\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, we are asked to evaluate the following double summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"395.5\" height=\"46\" style=\"width: 395.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIn which, in Matlab, evaluates to:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 754.578125px 30px; transform-origin: 754.578125px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  S =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e      5.5833\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePlease round-off \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to the nearest whole number. Therefore, the final output for the above example is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37.5\" height=\"18\" style=\"width: 37.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x)\r\n    s = round(sum(arrayfun(@(j) sum(arrayfun(@(i) sum(1./prod(nchoosek(1:j,i)')),2:j-1)),3:x)));\r\nend","test_suite":"%%\r\nx = 1:10;\r\ns_correct = [0 0 1 3 6 9 14 19 25 32];\r\nassert(isequal(arrayfun(@S,x),s_correct))\r\n%%\r\nx = 15;\r\ns_correct = 81;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 20;\r\ns_correct = 154;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 25:25:500;\r\ns_correct = [250 1095 2552 4625 7318 10630 14563 19118 24294 30093 36514 43558 51225 59515 68428 77965 88125 98908 110315 122346];\r\nassert(isequal(arrayfun(@S,x),s_correct))\r\n%%\r\nx = 1234;\r\ns_correct = 753724;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 12345;\r\ns_correct = 76094591;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 123456;\r\ns_correct = 7619358524;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1234567;\r\ns_correct = 762061661646;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1001001:1002001;\r\ns = arrayfun(@S,x);\r\nss = floor([mean(s) median(s) mode(s) std(s)])\r\nss_correct = [501489254654 501489212905 500988594352 289538048];\r\ns = s(floor(logspace(0,log10(length(x)),10)));\r\ns_correct = [500988594352 500989595339 500991597317 500997603276 501008614294 501033639784 501087696976 501202828490 501452158566 501990081457];\r\nassert(isequal(s,s_correct))\r\nassert(isequal(ss,ss_correct))\r\n%%\r\nfiletext = fileread('S.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'java') || contains(filetext, 'py') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-01-07T06:32:38.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2023-01-05T10:59:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-05T01:20:27.000Z","updated_at":"2026-01-22T20:49:35.000Z","published_at":"2023-01-05T10:36:53.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven two positive integers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\\\\ge b\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we define the function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR(a,b)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the sum of the reciprocals of the products of all combinations of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e taken \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at a time. In Matlab we say that:    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eR = sum(1./prod(nchoosek(a,b)'))\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. For example if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea=5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb=3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e R = sum(1./prod(nchoosek(1:5,3)'))\\n  R =\\n      0.7083]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, we are asked to evaluate the following double summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(x) = \\\\sum_{n=3}^{x} \\\\sum_{k=2}^{n-1} R(n,k).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(5) = \\\\sum_{n=3}^{5} \\\\sum_{k=2}^{n-1} R(n,k) = \\\\sum_{k=2}^2 R(3,k)+\\\\sum_{k=2}^3 R(4,k)+\\\\sum_{k=2}^4 R(5,k).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn which, in Matlab, evaluates to:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\\n  S =\\n      5.5833]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease round-off \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to the nearest whole number. Therefore, the final output for the above example is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":1046,"title":"Add two numbers","description":"Calculate the sum of two numbers.\r\n\r\nExample\r\n\r\n input  = [2 3]\r\n output =  5","description_html":"\u003cp\u003eCalculate the sum of two numbers.\u003c/p\u003e\u003cp\u003eExample\u003c/p\u003e\u003cpre\u003e input  = [2 3]\r\n output =  5\u003c/pre\u003e","function_template":"function y = addup(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [2 3];\r\ny_correct = 5;\r\nassert(isequal(addup(x),y_correct))\r\n\r\n%%\r\nx = [-3 3];\r\ny_correct = 0;\r\nassert(isequal(addup(x),y_correct))\r\n\r\n%%\r\nx = [2.1 48];\r\ny_correct = 50.1;\r\nassert(isequal(addup(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":11,"comments_count":3,"created_by":6303,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":1487,"test_suite_updated_at":"2013-07-11T15:49:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-11-21T13:35:24.000Z","updated_at":"2026-04-13T21:34:31.000Z","published_at":"2012-11-21T13:35:30.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the sum of two numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ input  = [2 3]\\n output =  5]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":42450,"title":"Summation of array","description":"Given an array, Find the sum of all of the elements in it\r\n\r\nExamples:\r\n\r\nInput x = [1 2 3 5; 4 5 6 7];\r\n\r\n\r\nOutput y is 33\r\n","description_html":"\u003cp\u003eGiven an array, Find the sum of all of the elements in it\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cp\u003eInput x = [1 2 3 5; 4 5 6 7];\u003c/p\u003e\u003cp\u003eOutput y is 33\u003c/p\u003e","function_template":"function y = sum_array(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [9 7 10;\r\n    10 1 10;\r\n     2 3  2;\r\n    10 6 10];\r\ny_correct = 80;\r\nassert(isequal(sum_array(x),y_correct))\r\n\r\n%%\r\nx = [4 8 7 2 9;\r\n     6 9 8 3 6;\r\n     7 6 5 5 8;\r\n     5 4 1 6 7;\r\n     9 2 3 5 5];\r\ny_correct = 140;\r\nassert(isequal(sum_array(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":45298,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":374,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-07-07T20:06:58.000Z","updated_at":"2026-04-12T21:25:36.000Z","published_at":"2015-07-07T20:09:47.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven an array, Find the sum of all of the elements in it\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInput x = [1 2 3 5; 4 5 6 7];\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOutput y is 33\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":43100,"title":"Add Me To Death (Sum sum sum)","description":"Given certain vector,  sum of the input vector, get the sum of 2 adjacent numbers in the input, and sum of the output vector (which is the sum of the 2  adjacent nos)\r\nSum sum sum :)","description_html":"\u003cp\u003eGiven certain vector,  sum of the input vector, get the sum of 2 adjacent numbers in the input, and sum of the output vector (which is the sum of the 2  adjacent nos)\r\nSum sum sum :)\u003c/p\u003e","function_template":"function [a b c] = addMeToDeath(x)\r\n  \r\nend","test_suite":"%%\r\nx = [1 2 3 4 5];\r\nsum_input  = 15;\r\nsum_output = 24;\r\nvec_sum_adjacent = [3 5 7 9]\r\n[a b c] = addMeToDeath(x)\r\nassert(isequal(a,sum_input))\r\nassert(isequal(b,sum_output))\r\nassert(isequal(c,vec_sum_adjacent))\r\n%%\r\nx = [7 8 9];\r\nsum_input  = 24;\r\nsum_output = 32;\r\nvec_sum_adjacent = [15 17]\r\n[a b c] = addMeToDeath(x)\r\nassert(isequal(a,sum_input))\r\nassert(isequal(b,sum_output))\r\nassert(isequal(c,vec_sum_adjacent))","published":true,"deleted":false,"likes_count":8,"comments_count":0,"created_by":13865,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":74,"test_suite_updated_at":"2016-10-29T17:05:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2016-10-06T07:20:13.000Z","updated_at":"2026-04-09T21:23:28.000Z","published_at":"2016-10-06T07:20:13.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven certain vector, sum of the input vector, get the sum of 2 adjacent numbers in the input, and sum of the output vector (which is the sum of the 2 adjacent nos) Sum sum sum :)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":730,"title":"How many trades represent all the profit?","description":"Given a list of results from trades made:\r\n\r\n[1 3 -4 2 -1 2 3]\r\n\r\nWe can add them up to see this series of trades made a profit of 6.  \r\n\r\nSince the two biggest trades sum to that, we can say that all the profit came from two trades.\r\n\r\nWhat is the minimum number of trades that represent the total profit (possibly exceeding it).  All trade lists will represent profitable series.","description_html":"\u003cp\u003eGiven a list of results from trades made:\u003c/p\u003e\u003cp\u003e[1 3 -4 2 -1 2 3]\u003c/p\u003e\u003cp\u003eWe can add them up to see this series of trades made a profit of 6.\u003c/p\u003e\u003cp\u003eSince the two biggest trades sum to that, we can say that all the profit came from two trades.\u003c/p\u003e\u003cp\u003eWhat is the minimum number of trades that represent the total profit (possibly exceeding it).  All trade lists will represent profitable series.\u003c/p\u003e","function_template":"function y = trade_profit(x)\r\n  y = x;\r\nend","test_suite":"%%\r\ntrades = [1 3 -4 2 -1 2 3]\r\ny_correct = 2;\r\nassert(isequal(trade_profit(trades),y_correct))\r\n\r\n%%\r\ntrades = [1 2 3 -5]\r\ny_correct = 1;\r\nassert(isequal(trade_profit(trades),y_correct))\r\n\r\n%%\r\ntrades = [1 2 3 4 5 6]\r\ny_correct = 6;\r\nassert(isequal(trade_profit(trades),y_correct))\r\n\r\n%%\r\ntrades = [-2 3 -4 5 -6 1 2 3 4 5]\r\ny_correct = 3;\r\nassert(isequal(trade_profit(trades),y_correct))","published":true,"deleted":false,"likes_count":12,"comments_count":2,"created_by":240,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":617,"test_suite_updated_at":"2012-06-11T13:22:29.000Z","rescore_all_solutions":false,"group_id":6,"created_at":"2012-06-01T13:38:13.000Z","updated_at":"2026-04-05T02:56:12.000Z","published_at":"2012-06-08T19:08:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a list of results from trades made:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e[1 3 -4 2 -1 2 3]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe can add them up to see this series of trades made a profit of 6.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince the two biggest trades sum to that, we can say that all the profit came from two trades.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the minimum number of trades that represent the total profit (possibly exceeding it). All trade lists will represent profitable series.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":660,"title":"Find a subset that divides the vector into equal halves","description":"Given a vector x, return the indices to elements that will sum to exactly half of the sum of all elements.\r\n\r\nExample: \r\n\r\n Input  x  = [1 2 3 4 5 6 7]\r\n Output xi = [1 6 7]\r\n\r\nbecause \r\n\r\n sum(x) = 28\r\n sum(x([1 6 7])) = 14\r\n\r\nThe answer is not necessarily unique and the order is unimportant. We will just test to make sure that |sum(x)/2| is |sum(x(xi))|\r\n","description_html":"\u003cp\u003eGiven a vector x, return the indices to elements that will sum to exactly half of the sum of all elements.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e Input  x  = [1 2 3 4 5 6 7]\r\n Output xi = [1 6 7]\u003c/pre\u003e\u003cp\u003ebecause\u003c/p\u003e\u003cpre\u003e sum(x) = 28\r\n sum(x([1 6 7])) = 14\u003c/pre\u003e\u003cp\u003eThe answer is not necessarily unique and the order is unimportant. We will just test to make sure that \u003ctt\u003esum(x)/2\u003c/tt\u003e is \u003ctt\u003esum(x(xi))\u003c/tt\u003e\u003c/p\u003e","function_template":"function xi = split_it(x)\r\n  xi = [];\r\nend","test_suite":"%%\r\nx = [1 2 3 4 5 6 7];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%%\r\nx = [2 2 2 2 2 2];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%%\r\nx = [2     5     4     5     4];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%% \r\nx = [1     3     1     1     9     7];\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n\r\n%% \r\nx = primes(100);\r\nxi = split_it(x);\r\nassert(isequal(sum(x(xi)),sum(x)/2));\r\n","published":true,"deleted":false,"likes_count":14,"comments_count":14,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":401,"test_suite_updated_at":"2013-11-06T16:04:24.000Z","rescore_all_solutions":false,"group_id":6,"created_at":"2012-05-04T19:18:34.000Z","updated_at":"2026-02-19T13:17:58.000Z","published_at":"2012-06-08T19:08:22.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a vector x, return the indices to elements that will sum to exactly half of the sum of all elements.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  x  = [1 2 3 4 5 6 7]\\n Output xi = [1 6 7]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ebecause\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ sum(x) = 28\\n sum(x([1 6 7])) = 14]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe answer is not necessarily unique and the order is unimportant. We will just test to make sure that\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum(x)/2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum(x(xi))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":58274,"title":"Easy Sequences 116: Summation involving nthroot, floor, and ceiling functions","description":"Given a postive integers  and , we are asked to evaluate the following summation:\r\n                            \r\n                            where: the symbol \"\" and \"\" are the floor and ceil functions, respectively.\r\nFor example, if  and , we have:\r\n\u003e\u003e  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\r\n\u003e\u003e  s = S(50,3)\r\n\u003e\u003e  s =\r\n        279","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 226px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 113px; transform-origin: 407px 113px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a postive integers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, we are asked to evaluate the following summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                            \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"266.5\" height=\"46\" style=\"width: 266.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                            where: the symbol \"\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23.5\" height=\"19\" style=\"width: 23.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\" and \"\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23.5\" height=\"19\" style=\"width: 23.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\" are the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/matlab/ref/floor.html#\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003efloor\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/matlab/ref/ceil.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003eceil\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e functions, respectively.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44\" height=\"18\" style=\"width: 44px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40px; transform-origin: 404px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt;  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt;  s = S(50,3)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u0026gt;\u0026gt;  s =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        279\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x,n)\r\n    s = sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\r\nend","test_suite":"%%\r\nx = 50; n = 3;\r\ns_correct = 279;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 100; n = 2;\r\ns_correct = 1339;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 1000:2000; n = 4;\r\ns = sum(arrayfun(@(i) S(i,n),x));\r\ns_correct = 15056572;\r\nassert(isequal(s,s_correct))\r\n%%\r\nx = 1234; n = 5;\r\ns_correct = 8508;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 10000:20000; n = 6;\r\ns = sum(arrayfun(@(i) S(i,n),x));\r\ns_correct = 1271494600;\r\nassert(isequal(s,s_correct))\r\n%%\r\nx = 123456; n = 7;\r\ns_correct = 1164369;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 100000:200000; n = 8;\r\ns = arrayfun(@(i) S(i,n),x);\r\nss = floor([mean(s),median(s),mode(s),std(s)])\r\nss_correct = [1205294 1205294 755294 259811];\r\nassert(isequal(ss,ss_correct))\r\n%%\r\nx = 1234567; n = 89;\r\ns_correct = 3703698;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 123456789; n = 10;\r\ns_correct = 1462257359;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 1234567891011; n = 12;\r\ns_correct = 23191068638973;\r\nassert(isequal(S(x,n),s_correct))\r\n%%\r\nx = 11111111111111; n = 111;\r\ns_correct = 33333333333330;\r\nassert(isequal(S(x,n),s_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-05-08T15:29:34.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-05-08T10:48:38.000Z","updated_at":"2023-05-08T15:29:34.000Z","published_at":"2023-05-08T15:28:45.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a postive integers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, we are asked to evaluate the following summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                            \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es= S(x,n) = \\\\sum_{i=1}^x \\\\bigg(\\\\Big\\\\lceil \\\\sqrt[n]{i -1} \\\\Big\\\\rceil+\\\\Big\\\\lfloor \\\\sqrt[n]{i+1} \\\\Big\\\\rfloor\\\\bigg)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                            where: the symbol \\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lfloor\\\\ \\\\rfloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e\\\" and \\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lceil\\\\ \\\\rceil\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e\\\" are the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/ref/floor.html#\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efloor\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/ref/ceil.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eceil\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e functions, respectively.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=50\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[\u003e\u003e  S = @(x,n) sum(arrayfun(@(i) ceil(nthroot(i-1,n))+floor(nthroot(i+1,n)),1:x));\\n\u003e\u003e  s = S(50,3)\\n\u003e\u003e  s =\\n        279]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57512,"title":"Easy Sequences 89: Double Summation of a Trigonometric Product","description":"Given and angle  in radians and a positive integer , evaluate the following product summation:\r\n                    \r\nwhich 'directly' translates to Matlab as:\r\n    \u003e\u003e PS = @(A,x) sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));\r\nFor example  and :\r\n    \u003e\u003e x = 3; A = 1;\r\n    \u003e\u003e PS(A,x)\r\n    ans =\r\n        8.9683\r\nPlease present your answer rounded-off to nearest 4 decimal places.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 288px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 492.078125px 144px; transform-origin: 492.078125px 144px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven and angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e in radians and a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, evaluate the following product summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e                    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"236\" height=\"46\" style=\"width: 236px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhich 'directly' translates to Matlab as:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; PS = @(A,x) sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 489.078125px 40px; transform-origin: 489.078125px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; x = 3; A = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; PS(A,x)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 489.078125px 10px; transform-origin: 489.078125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        8.9683\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ePlease present your answer rounded-off to nearest 4 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = PS(A,x)\r\n    p = sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));\r\nend","test_suite":"%%\r\nA = 1; x = 3;\r\np_correct = 8.9683;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 2:2:20; x = 3:3:30;\r\np_correct = [1.8697 -3.3674 -62.4997 11.5088 -4.8523 -71.5196 25.1022 -3.2420 -64.7730 49.3233];\r\nassert(isequal(arrayfun(@(i) PS(A(i),x(i)),1:10),p_correct))\r\n%%\r\np_correct = 4166.0746;\r\nassert(isequal(sum(arrayfun(@(i) PS(i,i),1:100)),p_correct))\r\n%%\r\nA = 100; x = 200;\r\np_correct = -739.8950;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 2000; x = 3000;\r\np_correct = 2041.1906;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 10000; x = 20000;\r\np_correct = -3131.6478;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 200000; x = 300000;\r\np_correct = -8386548.7261;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 1000000; x = 2000000;\r\np_correct = -11067350.2876;\r\nassert(isequal(PS(A,x),p_correct))\r\n%%\r\nA = 123456; xs = 200001:200100;\r\nps = arrayfun(@(x) PS(A,x),xs);\r\nss_correct = [-88558 -88558 -88580 12]; \r\nassert(isequal(floor([mean(ps) median(ps) mode(ps) std(ps)]),ss_correct))\r\n%%\r\nfiletext = fileread('PS.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'java') || contains(filetext, 'py') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-01-07T06:37:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-06T09:42:25.000Z","updated_at":"2025-11-22T20:00:54.000Z","published_at":"2023-01-07T06:37:48.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven and angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e in radians and a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, evaluate the following product summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e                    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ePS(A,x) = \\\\sum_{n=1}^{x} \\\\sum_{m=1}^{n} \\\\prod_{k=0}^{m-1} 2 \\\\sin \\\\left( \\\\frac_{k\\\\pi}^{m} +A \\\\right).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich 'directly' translates to Matlab as:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e PS = @(A,x) sum(arrayfun(@(n) sum(arrayfun(@(m) prod(arrayfun(@(k) 2*sin(k*pi/m+A),0:m-1)),1:n)),1:x));]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e x = 3; A = 1;\\n    \u003e\u003e PS(A,x)\\n    ans =\\n        8.9683]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePlease present your answer rounded-off to nearest 4 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57993,"title":"Easy Sequences 110: Integration of the Sum of a Recursive Trigonometric Function","description":"A trigonometric function, , is defined as follows:\r\n                ,  in radians\r\nApplying  recursively we define another function , for integer :\r\n                \r\nWe then define  as the sum of value of  from  to :\r\n                \r\nFinally, we are asked to evaluate the integral of  with respect to , over the real range :\r\n                \r\nFor example for , , , we have:\r\n  \u003e\u003e a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\r\n       a = 7.05797686912156\r\nPlease present the final output rounded-off to 6 decimal places. Therefore the final answer is .\r\n-------------------------\r\nNOTE: There are a number of ways to do numerical Integration in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the integral function shown above.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 507px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 462.578125px 253.5px; transform-origin: 462.578125px 253.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA trigonometric function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"31.5\" height=\"19\" style=\"width: 31.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, is defined as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"137.5\" height=\"19\" style=\"width: 137.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in radians\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eApplying \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAAmCAYAAABzhkOMAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAP6ADAAQAAAABAAAAJgAAAADyFT37AAAEHElEQVRoBe2YWYhOYRjHxzb2fRuyZi03lJLCBRqFcWEpDZEQJReWuxG5kISSubDFlRsllKUkM1nChaUpW7LUZCt7ZN9+f877dead95zvnOM7QzlP/b/vfZ/n+T/v9rzL9xUVZZLNQDYD2QxkM5DqDJQT/T5oW6BWrhJnddxYTS1CCfXeli5u9S2E2yGkZdgqwWIg30LISoKcBD3AqqQBKyD++ENcCml8FLbPYEOIT1LTXIjfweyoAeyVb+0jnqVcDV6BD2AWmAgkGsBy0Ax0BMPAdFAM/DGo5kR+B8BxsCanLVxhP6GGgD2gBtwCsWQb3lp5V+ps9Wyy3wG2jEShSblnG7z6Eb6/gAEB9kKoWxDkMbgJ2sQNqFnTQdTIQcw3eFF2gacObjk6TZripy3KSLW1Pl9DjS2HVtRPAZGTyAlILR3EhZ4ujb1uN7cbxXMwH7gW0fbP1YdS6pmr1S1EWXmlndLfL7o9voHrfmXKZZ0tWsDxYe3YK68rSnsmqXyEeNkiz6Gudk5b+rCqsqcMrAUVwN6/k9BtBtOAS6o85QKXMYkuysq74p5BqVUI6qjN2YRC97+yRTxhCzDi78c7lNqqtgxGId4j25C07m/0TowgD/BVRwZG5KjjWuleQIMT11xbKzzdPr51MOt8con2+legCbSvc5d/Xl2SwasTn4AGoHs+rhyGIK5QCrStlPJR5BlO4vUJcrb3fJBfUn1XiMVAK/A6QZCzPo6er5VA31HkheekDHJK2oPv4bWqgWsV4soFH6GW8jpfPV/RTHZgxqU9eO1Zia7AJHINklJdcgW8/1WK9mHeG9r7Tkl78Oa01Xu/ubMH4UodVmYCR4e71rN29jQP61k8RdqD16rp4JGYzvyuRfvUFdfFcy3hO+qNIYrh/bXBqxOm8aCXo3xcMhXlErDDZxzrK4cVtc+VacqaN0GOcVbev2/1UzaqmDs6ascVtzvYC7TyeuGZw9LEUPs7QSfgknGessZlTKI7BEmdEPQCawKiyGScxDka4twU2xQwCGiSz4OLQHqJBqEYT4B0GngV0DvCJdtRyn+pyxhHp/QpA9q/CmiwgXIHkE80SY+B0s8MxuboV5+Jq4nVOdHX57TRZ1cs3eF+u8/1V/EGn59AUGbY/s66/p35DkzHXN8Hncy6Sr3XxZ1XV52r6V8iE/slZftk749Og5GP/lky6U+xnoxBIz9l6j8h/eiFVv4uCFr9UdhKgeuHCuqibkAZqFdjmFRjVJaOCHNqaNsMGtSKLEqx4QleG8tSbCNx6K0wtaeHJ44QTNRVWguibMPgKClalPLnwCMQ+IMjQftt4FwF90D7BPwGo+jlpR8sur50ExRCjhFE/zEMLESwtGMU08BMEOeRFdYnnSftwhwyWzYD2QxkM/Bfz8BPP8zkcVJh+/8AAAAASUVORK5CYII=\" width=\"31.5\" height=\"19\" style=\"width: 31.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e recursively we define another function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"46\" height=\"19\" style=\"width: 46px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, for integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 51px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 25.5px; text-align: left; transform-origin: 384px 25.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-20px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"166.5\" height=\"51\" style=\"width: 166.5px; height: 51px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWe then define \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44\" height=\"19\" style=\"width: 44px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the sum of value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eR\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003e1\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"129.5\" height=\"46\" style=\"width: 129.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFinally, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ewe are asked to evaluate the integral of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e with respect to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, over the real range \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-21px\"\u003e\u003cimg 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width=\"204\" height=\"48\" style=\"width: 204px; height: 48px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42.5\" height=\"20\" style=\"width: 42.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"50\" height=\"20\" style=\"width: 50px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 40px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 459.578125px 20px; transform-origin: 459.578125px 20px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 459.578125px 10px; transform-origin: 459.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 459.578125px 10px; transform-origin: 459.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e       a = 7.05797686912156\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"85.5\" height=\"18\" style=\"width: 85.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e-------------------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThere are a number of ways to do \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/help/matlab/numerical-integration-and-differentiation.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumerical Integration\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003eintegral\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e function shown above.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function a = A(n,x1,x2) \r\n    y = x;\r\nend","test_suite":"%%\r\n[n,x1,x2] = deal(1:10,pi,2*pi);\r\na_correct = [3.065357 5.25927 7.057977 8.618935 10.016842 11.294017 12.477158 13.584381 14.628646 15.619599];\r\nassert(all(abs(arrayfun(@(i) A(i,x1,x2),n)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(12,exp(1),exp(1.5));\r\na_correct = 9.752678;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(15,-2,10);\r\na_correct = 50.909769;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(25,-10*pi,0);\r\na_correct = -267.631308;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(1000,0,1000);\r\na_correct = 61793.524569;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(10000,5,1234);\r\na_correct = 244011.112390;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(123456,10000,12345);\r\na_correct = 1644471.557504;\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\n[n,x1,x2] = deal(1:1000,-pi,20*exp(1));\r\na = arrayfun(@(i) A(i,x1,x2),n);\r\ns = round([sum(a) sum(diff(a,1)) sum(diff(a,2)) sum(diff(a,3))]);\r\ns_correct = [2087798 3114 -35 7];\r\nassert(isequal(s,s_correct))\r\n%%\r\n[n,x1,x2] = deal(randi(15),rand(),100*rand());\r\nt = 'sin(atan(';\r\na_correct = 0;\r\nfor i = 1:n\r\n    a_correct = a_correct + integral(str2num(['@(x)' repmat('sin(atan(',1,i) 'x' repmat('))',1,i)]),x1,x2);\r\nend\r\na_correct = round(a_correct,6);\r\nassert(all(abs(A(n,x1,x2)-a_correct)\u003c=0.000001))\r\n%%\r\nfiletext = fileread('A.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'assignin') || contains(filetext, 'evalin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-04-23T09:34:40.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-04-16T08:49:13.000Z","updated_at":"2023-04-23T09:34:40.000Z","published_at":"2023-04-16T17:57:39.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA trigonometric function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is defined as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x) = \\\\sin(\\\\arctan(x))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in radians\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eApplying \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e recursively we define another function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{R}(x,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{R}(x,n)=\\\\underbrace{\\\\text{T}(\\\\text{T}(\\\\text{T}(...\\\\text{T}(x))))}\\\\\\\\_{\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\text{n times}}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe then define \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{S}(x,n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the sum of value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{R}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{S}(x,n) = \\\\sum_{k=1}^{n} \\\\text{R}(x,k) \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFinally, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ewe are asked to evaluate the integral of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{S}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e with respect to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, over the real range \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e[x_1,x_2]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea=\\\\text{A}(n,x_1,x_2) = \\\\int^{x_2}_{x_1} \\\\text{S}(x,n)\\\\ \\\\mathrm{d}x}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1=\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2=2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)\\n       a = 7.05797686912156]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea=7.057977\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-------------------------\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eThere are a number of ways to do \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/help/matlab/numerical-integration-and-differentiation.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumerical Integration\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e in Matlab.  Just make sure that the output would be accurate within 6 decimal places of the value obtained using the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eintegral\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e function shown above.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57983,"title":"Easy Sequences 109: Summation of Derivatives of a Trigonometric Function","description":"A trigonometric function, ,  is defined as follows:\r\n                \r\n                where:     ; and \r\n                                 is in radians.\r\nIn this problem we are asked to evaluate the following summation:\r\n                \r\n                where:     is a real number; \r\n                                 is an integer; \r\n                                 or the -th derivative of  with respect to ; and \r\n                                .\r\nFor example for  and , we have:\r\n  \u003e\u003e  syms x;\r\n  \u003e\u003e  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\r\n  \u003e\u003e  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\r\n  \u003e\u003e  s = vpa(subs(S,x,2))\r\n        s = 0.10315887444431633673347091141408;\r\nPlease present the final output rounded-off to 6 decimal places. Therefore the final answer is .\r\n-------------------------\r\nNOTE: Symbolic toolbox is not available to Cody players. It is possible to do numerical differentiation in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using syms, as shown above.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 622px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 311px; transform-origin: 407px 311px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA trigonometric function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"31.5\" height=\"19\" style=\"width: 31.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,  is defined as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 38px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19px; text-align: left; transform-origin: 384px 19px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-14px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"133.5\" height=\"38\" style=\"width: 133.5px; height: 38px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                where:     \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"152.5\" height=\"20\" style=\"width: 152.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is in radians.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eIn this problem we are asked to evaluate the following summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"142.5\" height=\"46\" style=\"width: 142.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                where:    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is a real number; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is an integer; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"112.5\" height=\"37\" style=\"width: 112.5px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e-th derivative of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with respect to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; and \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"91\" height=\"20\" style=\"width: 91px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFor example for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAASaADAAQAAAABAAAAJAAAAABLVRfgAAAC+ElEQVRoBe2X24tPURTHZ9xikkmIB5fkQSKaQhlFjXFtPHhRaIqS8oL/YF4oKeRNgxqXp1E8ICHSeJB75IVxSRnJJbnmNvh8OUu703Z++/cz9bvtVZ/Z++y19vnt9T3nrL2npiZaVCAqEBWICkQFKk2B2kpLKEc+k/A3wnQYDFfgMjyEqreBKNAGX+Gnhw7G+kNV2z6y94njjh0nZlC1qrQwEegu7QLQZ1YPi+AOuEK1cF2Vdoasn8JoT/ZDGHsAJtRuT0zFDw0lwy+wKiPTVnwm0s2MuIp1aRc7B1lFeRp+E+m6T4kBvsFkbCRtQ0I3rQqbbAIsB22nJ+E8lKqp5qgmZdlnx/nI6Wd2d+LVN2zqql2TzFhH+yHlm5X4yrVpdvJZHZrEZAJXgp0pvtEfDpvgO6i4nQYTcSP9UDtG4Os+QLtUX9kebqRcnoN2vmDT4es9aPIFWAGfQNumbDOYSEt+j4T9UX2wef/T5vObWSurw/kiWZNeDK/9qybNIVo7g0zf9UHYAGdBpoJodts6Ae12Yo4GxOUKuZErINDfRtwo6EwInPYnbCuNPekn9NtTs+8lfh3QytVmsnCVkqug81Lepokm0n36dc4dxjs+FflytLEsugf0j+2YQhIYwaReMJGaUjdZ7/gWp3zlcDmMRd4CiTQxZMG+mtTMxH7JZNUgFW7X7Nyh80WX6wjoa3ebHxCXK0RFtpDzmT6rEzAO5sFjKMgOMMveoqWpO0i8V4nfingqJPOymLubduxT8A6yznZr8c+Gv+Z7k2ybf0mUknKtgQt9jjITSaIehouQy4q1u+nhHoEm0IO/Bj7T27UXZvicNjaFjr1F7TbotFscvwTbBt1QD6VqtSxsPyivZ9Dh4RBjqlO90AWZ5orQ4oncwZiJeIn+W3DPTJ4pRR/axQpszSFta64VdyY3/EirIpe2ZQz8AP1YDzRCKdtcFhcijMW8Id6Xd945TmWGTuR9crO8fz1OiApEBaICUYGoQFQgKhAVKBkFfgH9tfMUfn5wuwAAAABJRU5ErkJggg==\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 100px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 50px; transform-origin: 404px 50px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  syms x;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt;  s = vpa(subs(S,x,2))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10px; transform-origin: 404px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        s = 0.10315887444431633673347091141408;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"83.5\" height=\"18\" style=\"width: 83.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e-------------------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSymbolic toolbox is not available to Cody players. It is possible to do \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.unioviedo.es/compnum/labs/lab07_der_int/lab07_der_int.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-weight: 700; \"\u003enumerical differentiation\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003esyms\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, as shown above.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x,n)\r\n  s = x;\r\nend","test_suite":"%%\r\nx = 1:10; n = 2;\r\ns_correct = [-1.325444 0.103159 1.239586 -1.134858 -0.29505 1.380427 -0.85387 -0.669756 1.411304 -0.504863];\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 11:20; n = 2*x;\r\ns_correct = [-0.991110 -0.481399 -0.115639 -0.691700 1.142283 1.385650 -1.377653 -1.119743 0.658705  0.078175];\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 25.25; n = 25;\r\ns_correct = 1.945253;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 100; n = 100;\r\ns_correct = -0.386110;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 0.000123; n = 123;\r\ns_correct = 0.000001;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 1234; n = 1:1000;\r\na_correct = [275 250];\r\nassert(isequal([round(sum(S(x,n))) sum(round(S(x,n)))],a_correct))\r\n%%\r\nx = 123456; n = 123456;\r\ns_correct = 0.899338;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = 123456789.10111213; n = 123456789;\r\ns_correct = -1.993727;\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nx = rand()*randi(1000); n = randi(10)-1;\r\nTs = { \r\n       @(a) (2*tan(a) - tan(a)^2 + 1)/(tan(a)^2 + 1);\r\n       @(a) (2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2)/(tan(a)^2 + 1) - (2*tan(a)*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1);\r\n       @(a) 2*tan(a)^2 - 4*tan(a) - (4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2)/(tan(a)^2 + 1) + (4*tan(a)^2*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (4*tan(a)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - 2;\r\n       @(a) 8*tan(a)*(2*tan(a) - tan(a)^2 + 1) - 12*tan(a)^2 - (16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2)/(tan(a)^2 + 1) + 12*tan(a)*(tan(a)^2 + 1) + (12*tan(a)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (8*tan(a)^3*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (6*tan(a)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - 12;\r\n       @(a) 8*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) - (16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3)/(tan(a)^2 + 1) + 48*tan(a)^2*(tan(a)^2 + 1) - 24*tan(a)^2*(2*tan(a) - tan(a)^2 + 1) - 48*tan(a)*(tan(a)^2 + 1) + 32*tan(a)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 24*(tan(a)^2 + 1)^2 - (24*tan(a)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (32*tan(a)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + (8*tan(a)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (16*tan(a)^4*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1);\r\n       @(a) 320*tan(a)*(tan(a)^2 + 1)^2 - 160*tan(a)^2*(tan(a)^2 + 1) + 160*tan(a)^3*(tan(a)^2 + 1) - 120*tan(a)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - (272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3)/(tan(a)^2 + 1) + 40*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 64*tan(a)^3*(2*tan(a) - tan(a)^2 + 1) - 80*tan(a)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 80*(tan(a)^2 + 1)^2 + (80*tan(a)^3*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (80*tan(a)^4*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (32*tan(a)^5*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (40*tan(a)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (10*tan(a)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1);\r\n       @(a) 480*tan(a)^4*(tan(a)^2 + 1) - 960*tan(a)*(tan(a)^2 + 1)^2 - 480*tan(a)^3*(tan(a)^2 + 1) - (64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4)/(tan(a)^2 + 1) + 360*tan(a)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 384*tan(a)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 160*tan(a)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 32*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) + 2640*tan(a)^2*(tan(a)^2 + 1)^2 - 120*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 160*tan(a)^4*(2*tan(a) - tan(a)^2 + 1) + 480*(tan(a)^2 + 1)^3 - (240*tan(a)^4*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (60*tan(a)^2*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - (192*tan(a)^5*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + 128*tan(a)^2*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) + (12*tan(a)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 192*tan(a)*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (64*tan(a)^6*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (160*tan(a)^3*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1);\r\n       @(a) 11424*tan(a)*(tan(a)^2 + 1)^3 - 280*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 224*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 1344*tan(a)^4*(tan(a)^2 + 1) + 1344*tan(a)^5*(tan(a)^2 + 1) - 1344*tan(a)^3*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 1120*tan(a)^4*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 7392*tan(a)^2*(tan(a)^2 + 1)^2 + 17472*tan(a)^3*(tan(a)^2 + 1)^2 - (7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4)/(tan(a)^2 + 1) + 384*tan(a)^5*(2*tan(a) - tan(a)^2 + 1) + 840*tan(a)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 280*tan(a)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 1344*(tan(a)^2 + 1)^3 + (672*tan(a)^5*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (280*tan(a)^3*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) + (448*tan(a)^6*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + 128*tan(a)*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) - 384*tan(a)^3*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) + 672*tan(a)*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (128*tan(a)^7*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (560*tan(a)^4*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (84*tan(a)^2*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) + 896*tan(a)^2*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (14*tan(a)*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1);\r\n       @(a) 3584*tan(a)^6*(tan(a)^2 + 1) - 30464*tan(a)*(tan(a)^2 + 1)^3 - 3584*tan(a)^5*(tan(a)^2 + 1) - (256*tan(a)^8*(tan(a)^2 + 1) - 256*tan(a)^7*(tan(a)^2 + 1) - 15872*tan(a)*(tan(a)^2 + 1)^4 + 137216*tan(a)^2*(tan(a)^2 + 1)^4 - 49152*tan(a)^3*(tan(a)^2 + 1)^3 + 185856*tan(a)^4*(tan(a)^2 + 1)^3 - 15360*tan(a)^5*(tan(a)^2 + 1)^2 + 31616*tan(a)^6*(tan(a)^2 + 1)^2 + 7936*(tan(a)^2 + 1)^5)/(tan(a)^2 + 1) + 4480*tan(a)^4*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 1680*tan(a)^2*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) + 3072*tan(a)^5*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 448*tan(a)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) + 128*(tan(a)^2 + 1)^3*(2*tan(a) - tan(a)^2 + 1) + 161280*tan(a)^2*(tan(a)^2 + 1)^3 - 46592*tan(a)^3*(tan(a)^2 + 1)^2 + 102144*tan(a)^4*(tan(a)^2 + 1)^2 - 560*(tan(a)^2 + 1)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 896*tan(a)^6*(2*tan(a) - tan(a)^2 + 1) - 3584*tan(a)^3*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 15232*(tan(a)^2 + 1)^4 + 896*(tan(a)^2 + 1)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (1792*tan(a)^6*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (1120*tan(a)^4*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - (1024*tan(a)^7*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (112*tan(a)^2*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) + 1152*tan(a)^4*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) + (16*tan(a)*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) - 640*tan(a)^2*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) + (256*tan(a)^8*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (1792*tan(a)^5*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (448*tan(a)^3*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 3584*tan(a)^2*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 1024*tan(a)*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 3072*tan(a)^3*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 1792*tan(a)*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2);\r\n       @(a) 1152*(tan(a)^2 + 1)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 571392*tan(a)*(tan(a)^2 + 1)^4 - 9216*tan(a)^6*(tan(a)^2 + 1) + 9216*tan(a)^7*(tan(a)^2 + 1) - 1008*(tan(a)^2 + 1)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) - 13824*tan(a)^5*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - 8064*tan(a)^3*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 8064*tan(a)^6*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 2688*(tan(a)^2 + 1)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 414720*tan(a)^2*(tan(a)^2 + 1)^3 + 1769472*tan(a)^3*(tan(a)^2 + 1)^3 - 262656*tan(a)^4*(tan(a)^2 + 1)^2 + 552960*tan(a)^5*(tan(a)^2 + 1)^2 - (353792*tan(a)*(tan(a)^2 + 1)^5 - 512*tan(a)^8*(tan(a)^2 + 1) + 512*tan(a)^9*(tan(a)^2 + 1) - 274432*tan(a)^2*(tan(a)^2 + 1)^4 + 1841152*tan(a)^3*(tan(a)^2 + 1)^4 - 371712*tan(a)^4*(tan(a)^2 + 1)^3 + 1304832*tan(a)^5*(tan(a)^2 + 1)^3 - 63232*tan(a)^6*(tan(a)^2 + 1)^2 + 128512*tan(a)^7*(tan(a)^2 + 1)^2 - 15872*(tan(a)^2 + 1)^5)/(tan(a)^2 + 1) + 2048*tan(a)^7*(2*tan(a) - tan(a)^2 + 1) + 13440*tan(a)^4*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 3024*tan(a)^2*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) - 39168*(tan(a)^2 + 1)^4 - 672*tan(a)*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4) + (4608*tan(a)^7*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (4032*tan(a)^5*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (2304*tan(a)^8*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) + (672*tan(a)^3*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) - 512*tan(a)*(tan(a)^2 + 1)^3*(2*tan(a) - tan(a)^2 + 1) - 3072*tan(a)^5*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) - 10752*tan(a)^2*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 4032*tan(a)*(tan(a)^2 + 1)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) + 2048*tan(a)^3*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) - (512*tan(a)^9*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) - (5376*tan(a)^6*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (2016*tan(a)^4*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 4608*tan(a)*(tan(a)^2 + 1)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 13824*tan(a)^3*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (144*tan(a)^2*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) + 10368*tan(a)^4*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + (18*tan(a)*(256*tan(a)^8*(tan(a)^2 + 1) - 256*tan(a)^7*(tan(a)^2 + 1) - 15872*tan(a)*(tan(a)^2 + 1)^4 + 137216*tan(a)^2*(tan(a)^2 + 1)^4 - 49152*tan(a)^3*(tan(a)^2 + 1)^3 + 185856*tan(a)^4*(tan(a)^2 + 1)^3 - 15360*tan(a)^5*(tan(a)^2 + 1)^2 + 31616*tan(a)^6*(tan(a)^2 + 1)^2 + 7936*(tan(a)^2 + 1)^5))/(tan(a)^2 + 1);\r\n       @(a) 6720*(tan(a)^2 + 1)^2*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 1428480*tan(a)*(tan(a)^2 + 1)^4 - 23040*tan(a)^7*(tan(a)^2 + 1) + 23040*tan(a)^8*(tan(a)^2 + 1) + 40320*tan(a)^6*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (1024*tan(a)^10*(tan(a)^2 + 1) - 1024*tan(a)^9*(tan(a)^2 + 1) - 707584*tan(a)*(tan(a)^2 + 1)^5 + 9061376*tan(a)^2*(tan(a)^2 + 1)^5 - 3682304*tan(a)^3*(tan(a)^2 + 1)^4 + 21253376*tan(a)^4*(tan(a)^2 + 1)^4 - 2609664*tan(a)^5*(tan(a)^2 + 1)^3 + 8728576*tan(a)^6*(tan(a)^2 + 1)^3 - 257024*tan(a)^7*(tan(a)^2 + 1)^2 + 518656*tan(a)^8*(tan(a)^2 + 1)^2 + 353792*(tan(a)^2 + 1)^6)/(tan(a)^2 + 1) + 33600*tan(a)^4*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) + 20480*tan(a)^7*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) + 5040*tan(a)^2*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4) - 512*(tan(a)^2 + 1)^4*(2*tan(a) - tan(a)^2 + 1) - 960*tan(a)*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4) + 12349440*tan(a)^2*(tan(a)^2 + 1)^4 - 4423680*tan(a)^3*(tan(a)^2 + 1)^3 + 16727040*tan(a)^4*(tan(a)^2 + 1)^3 - 1382400*tan(a)^5*(tan(a)^2 + 1)^2 + 2845440*tan(a)^6*(tan(a)^2 + 1)^2 - 1680*(tan(a)^2 + 1)*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4) - 4608*tan(a)^8*(2*tan(a) - tan(a)^2 + 1) - 46080*tan(a)^5*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - 16128*tan(a)^3*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) + 714240*(tan(a)^2 + 1)^5 - 5760*(tan(a)^2 + 1)^3*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + 8064*tan(a)*(tan(a)^2 + 1)*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3) + 28800*tan(a)^2*(tan(a)^2 + 1)^2*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) - (11520*tan(a)^8*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) - (13440*tan(a)^6*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) + 20480*tan(a)^3*(tan(a)^2 + 1)^2*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - (5120*tan(a)^9*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2))/(tan(a)^2 + 1) - (3360*tan(a)^4*(64*tan(a)^6*(tan(a)^2 + 1) - 64*tan(a)^5*(tan(a)^2 + 1) - 544*tan(a)*(tan(a)^2 + 1)^3 + 2880*tan(a)^2*(tan(a)^2 + 1)^3 - 832*tan(a)^3*(tan(a)^2 + 1)^2 + 1824*tan(a)^4*(tan(a)^2 + 1)^2 + 272*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) + 8192*tan(a)^6*(tan(a)^2 + 1)*(2*tan(a) - tan(a)^2 + 1) - 15360*tan(a)*(tan(a)^2 + 1)^2*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) + 46080*tan(a)^3*(tan(a)^2 + 1)*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2) - (180*tan(a)^2*(256*tan(a)^8*(tan(a)^2 + 1) - 256*tan(a)^7*(tan(a)^2 + 1) - 15872*tan(a)*(tan(a)^2 + 1)^4 + 137216*tan(a)^2*(tan(a)^2 + 1)^4 - 49152*tan(a)^3*(tan(a)^2 + 1)^3 + 185856*tan(a)^4*(tan(a)^2 + 1)^3 - 15360*tan(a)^5*(tan(a)^2 + 1)^2 + 31616*tan(a)^6*(tan(a)^2 + 1)^2 + 7936*(tan(a)^2 + 1)^5))/(tan(a)^2 + 1) + (20*tan(a)*(353792*tan(a)*(tan(a)^2 + 1)^5 - 512*tan(a)^8*(tan(a)^2 + 1) + 512*tan(a)^9*(tan(a)^2 + 1) - 274432*tan(a)^2*(tan(a)^2 + 1)^4 + 1841152*tan(a)^3*(tan(a)^2 + 1)^4 - 371712*tan(a)^4*(tan(a)^2 + 1)^3 + 1304832*tan(a)^5*(tan(a)^2 + 1)^3 - 63232*tan(a)^6*(tan(a)^2 + 1)^2 + 128512*tan(a)^7*(tan(a)^2 + 1)^2 - 15872*(tan(a)^2 + 1)^5))/(tan(a)^2 + 1) + 3072*tan(a)^2*(tan(a)^2 + 1)^3*(2*tan(a) - tan(a)^2 + 1) - 7168*tan(a)^4*(tan(a)^2 + 1)^2*(2*tan(a) - tan(a)^2 + 1) + (1024*tan(a)^10*(2*tan(a) - tan(a)^2 + 1))/(tan(a)^2 + 1) + (15360*tan(a)^7*(16*tan(a)*(tan(a)^2 + 1)^2 - 8*tan(a)^2*(tan(a)^2 + 1) + 8*tan(a)^3*(tan(a)^2 + 1) - 4*(tan(a)^2 + 1)^2))/(tan(a)^2 + 1) + (8064*tan(a)^5*(272*tan(a)*(tan(a)^2 + 1)^3 - 32*tan(a)^4*(tan(a)^2 + 1) + 32*tan(a)^5*(tan(a)^2 + 1) - 176*tan(a)^2*(tan(a)^2 + 1)^2 + 416*tan(a)^3*(tan(a)^2 + 1)^2 - 32*(tan(a)^2 + 1)^3))/(tan(a)^2 + 1) - 51840*tan(a)^4*(tan(a)^2 + 1)*(4*tan(a)^2*(tan(a)^2 + 1) - 4*tan(a)*(tan(a)^2 + 1) + 2*(tan(a)^2 + 1)^2) + (960*tan(a)^3*(7936*tan(a)*(tan(a)^2 + 1)^4 - 128*tan(a)^6*(tan(a)^2 + 1) + 128*tan(a)^7*(tan(a)^2 + 1) - 5760*tan(a)^2*(tan(a)^2 + 1)^3 + 24576*tan(a)^3*(tan(a)^2 + 1)^3 - 3648*tan(a)^4*(tan(a)^2 + 1)^2 + 7680*tan(a)^5*(tan(a)^2 + 1)^2 - 544*(tan(a)^2 + 1)^4))/(tan(a)^2 + 1) - 26880*tan(a)^2*(tan(a)^2 + 1)*(16*tan(a)^4*(tan(a)^2 + 1) - 16*tan(a)^3*(tan(a)^2 + 1) - 32*tan(a)*(tan(a)^2 + 1)^2 + 88*tan(a)^2*(tan(a)^2 + 1)^2 + 16*(tan(a)^2 + 1)^3) - 5120*tan(a)*(tan(a)^2 + 1)^3*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2) - 30720*tan(a)^5*(tan(a)^2 + 1)*(2*tan(a)^2 - 2*tan(a)*(tan(a)^2 + 1) + 2);\r\n     };     \r\ns = S(x,n);\r\ns_correct = 0;\r\nfor  i = 1:n+1\r\n    s_correct = s_correct + Ts{i}(x) / 2^(i-1);\r\nend\r\ns_correct = round(s_correct,6);\r\nassert(all(abs(S(x,n)-s_correct)\u003c=0.000001))\r\n%%\r\nfiletext = fileread('S.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'assignin') || contains(filetext, 'evalin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-04-16T12:26:21.000Z","deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2023-04-15T15:36:04.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2023-04-14T18:29:33.000Z","updated_at":"2023-04-16T12:26:21.000Z","published_at":"2023-04-15T14:27:33.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA trigonometric function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,  is defined as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}(x)=\\\\frac^{1+2\\\\tan(x)-\\\\tan^2(x)}_{1+\\\\tan^2(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                where:     \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\tan^2(x) = \\\\tan(x)\\\\cdot\\\\tan(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is in radians.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eIn this problem we are asked to evaluate the following summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es=\\\\text{S}(x,n)=\\\\sum_{k=0}^n \\\\frac^{\\\\text{T}^{(k)}(x)}_{2^k}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                where:    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a real number; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is an integer; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}^{(k)}(x)=\\\\frac{\\\\mathrm{d}^k}{\\\\mathrm{d}x^k}\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e-th derivative of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with respect to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; and \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\text{T}^{(0)}(x)=\\\\text{T}(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en=2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e  syms x;\\n  \u003e\u003e  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);\\n  \u003e\u003e  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;\\n  \u003e\u003e  s = vpa(subs(S,x,2))\\n        s = 0.10315887444431633673347091141408;]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease present the final output rounded-off to 6 decimal places. Therefore the final answer is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es=0.103159\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-------------------------\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eSymbolic toolbox is not available to Cody players. It is possible to do \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.unioviedo.es/compnum/labs/lab07_der_int/lab07_der_int.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumerical differentiation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esyms\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, as shown above.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8071,"title":"summation of the reciprocals ","description":"Determine the summation of the reciprocals of numbers with length equal to N,in the meanwhile there is a digit K excluded from the numbers. \r\n\r\nreference:( \u003chttp://en.wikipedia.org/wiki/Kempner_series\u003e )\r\n\r\nFor example:\r\n\r\nif N = 2 and K = 7 \r\n\r\nthe matrix are as following:\r\n\r\n    10    20    30    40    50    60    80    90\r\n    11    21    31    41    51    61    81    91\r\n    12    22    32    42    52    62    82    92\r\n    13    23    33    43    53    63    83    93\r\n    14    24    34    44    54    64    84    94\r\n    15    25    35    45    55    65    85    95\r\n    16    26    36    46    56    66    86    96\r\n    18    28    38    48    58    68    88    98\r\n    19    29    39    49    59    69    89    99\r\n\r\nouput then can be the summation of the reciprocals of above matrix\r\n\r\noutput = 2.01554407485017\r\n\r\nSo, give the input N and K, determine the output as mentioned above  \r\n\r\n\r\n\r\n\r\n\r\n ","description_html":"\u003cp\u003eDetermine the summation of the reciprocals of numbers with length equal to N,in the meanwhile there is a digit K excluded from the numbers.\u003c/p\u003e\u003cp\u003ereference:( \u003ca href = \"http://en.wikipedia.org/wiki/Kempner_series\"\u003ehttp://en.wikipedia.org/wiki/Kempner_series\u003c/a\u003e )\u003c/p\u003e\u003cp\u003eFor example:\u003c/p\u003e\u003cp\u003eif N = 2 and K = 7\u003c/p\u003e\u003cp\u003ethe matrix are as following:\u003c/p\u003e\u003cpre\u003e    10    20    30    40    50    60    80    90\r\n    11    21    31    41    51    61    81    91\r\n    12    22    32    42    52    62    82    92\r\n    13    23    33    43    53    63    83    93\r\n    14    24    34    44    54    64    84    94\r\n    15    25    35    45    55    65    85    95\r\n    16    26    36    46    56    66    86    96\r\n    18    28    38    48    58    68    88    98\r\n    19    29    39    49    59    69    89    99\u003c/pre\u003e\u003cp\u003eouput then can be the summation of the reciprocals of above matrix\u003c/p\u003e\u003cp\u003eoutput = 2.01554407485017\u003c/p\u003e\u003cp\u003eSo, give the input N and K, determine the output as mentioned above\u003c/p\u003e","function_template":"function output = sum_reciprocals(N,K)\r\n  output  = [];\r\nend","test_suite":"%%\r\nN = 2;\r\nK = 7;\r\nassert(abs(sum_reciprocals(N,K)-2.01554407485016762)\u003c1e-8)\r\n%%\r\nN = 5;\r\nK = 7;\r\nassert(abs(sum_reciprocals(N,K)-1.440856832360981609753)\u003c1e-8)\r\n%%\r\nN = 8;\r\nK = 9;\r\nassert(abs(sum_reciprocals(N,K)-1.0714523172876426748)\u003c1e-8)\r\n%%\r\nN = 2:8;\r\nK = 9;\r\nM = arrayfun(@(x)sum_reciprocals(x,K),N);\r\nR = spline(N,M,[3.3 5.93]);\r\nassert(all(abs(R-[1.7588093884770861, 1.3325666476222477463])\u003c1e-8))","published":true,"deleted":false,"likes_count":3,"comments_count":1,"created_by":13709,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":"2015-04-08T14:05:58.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2015-04-08T11:21:46.000Z","updated_at":"2015-04-10T00:50:25.000Z","published_at":"2015-04-08T11:21:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the summation of the reciprocals of numbers with length equal to N,in the meanwhile there is a digit K excluded from the numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ereference:(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Kempner_series\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Kempner_series\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e )\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eif N = 2 and K = 7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe matrix are as following:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    10    20    30    40    50    60    80    90\\n    11    21    31    41    51    61    81    91\\n    12    22    32    42    52    62    82    92\\n    13    23    33    43    53    63    83    93\\n    14    24    34    44    54    64    84    94\\n    15    25    35    45    55    65    85    95\\n    16    26    36    46    56    66    86    96\\n    18    28    38    48    58    68    88    98\\n    19    29    39    49    59    69    89    99]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eouput then can be the summation of the reciprocals of above matrix\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eoutput = 2.01554407485017\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSo, give the input N and K, determine the output as mentioned above\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":57517,"title":"Easy Sequences 90: Triple Summation of a Combinatorial Function","description":"Given a positive integer , we are asked to evaluate the following triple summation:\r\n                \r\nThe symbol  is the combination function, nchoosek in Matlab. Therefore, we can write the function in Matlab as follows:\r\n    \u003e\u003e S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));\r\nHence, for , we have:\r\n    \u003e\u003e S(5)\r\n    \u003e\u003e ans =\r\n       -1.3302\r\nPlease round-off the output to the nearest 4 decimal places.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 313px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 563.71875px 156.5px; transform-origin: 563.71875px 156.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, we are asked to evaluate the following triple summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 57px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 28.5px; text-align: left; transform-origin: 384px 28.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; 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\" width=\"169\" height=\"57\" style=\"width: 169px; height: 57px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 55px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 27.5px; text-align: left; transform-origin: 384px 27.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe symbol \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-14px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"34\" style=\"width: 28.5px; height: 34px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the combination function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; font-weight: 700; \"\u003enchoosek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in Matlab. Therefore, we can write the function in Matlab as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eHence, for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we have:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 560.71875px 30px; transform-origin: 560.71875px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; S(5)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u0026gt;\u0026gt; ans =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e       -1.3302\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ePlease round-off the output to the nearest 4 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x)\r\n    s = sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));\r\nend","test_suite":"%%\r\nx = 1:20;\r\ns_correct = [1.0000 0.6667 0.1333 -0.5429 -1.3302 -2.2084 -3.1635 -4.1853 -5.2659 -6.3992 -7.5801 -8.8044 -10.0688 -11.3702 -12.706 -14.0742 -15.4726 -16.8996 -18.3536 -19.8333];\r\nassert(isequal(arrayfun(@S,x),s_correct))\r\n%%\r\nx = 25;\r\ns_correct = -27.5763;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 50;\r\ns_correct = -72.3576;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 100;\r\ns_correct = -179.0764;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1000;\r\ns_correct = -2936.8505;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 10000;\r\ns_correct = -40871.0455;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 100000;\r\ns_correct = -523824.1437;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1000000;\r\ns_correct = -6389513.2367;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nxs = 1000001:2000000;\r\ns = arrayfun(@(x) S(x),xs);\r\nss_correct = [-9902420 -9888371 -13472171 2045750];\r\nassert(isequal(floor([mean(s) median(s) mode(s) std(s)]),ss_correct))\r\n%%\r\nfiletext = fileread('S.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'java') || contains(filetext, 'py') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-01-07T10:58:56.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-07T09:14:32.000Z","updated_at":"2023-01-07T10:58:57.000Z","published_at":"2023-01-07T10:58:56.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, we are asked to evaluate the following triple summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(x) = \\\\sum_{y=0}^{x} \\\\sum_{n=0}^{y} \\\\sum_{k=0}^n \\\\frac_{(-4)^{k} {n \\\\choose k}}^{2k \\\\choose k}.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe symbol \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en \\\\choose k\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the combination function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enchoosek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr/\u003e\u003cw:t\u003e in Matlab. Therefore, we can write the function in Matlab as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHence, for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we have:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e S(5)\\n    \u003e\u003e ans =\\n       -1.3302]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ePlease round-off the output to the nearest 4 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57507,"title":"Easy Sequences 88: Double Summation of a Combinatorial Function","description":"Given two positive integers  and , , we define the function  as the sum of the reciprocals of the products of all combinations of  to  taken  at a time. In Matlab we say that:    R = sum(1./prod(nchoosek(a,b)')). For example if  and :\r\n  \u003e\u003e R = sum(1./prod(nchoosek(1:5,3)'))\r\n  R =\r\n      0.7083\r\nGiven a positive integer , we are asked to evaluate the following double summation:\r\n                \r\nFor example for :\r\n                \r\nIn which, in Matlab, evaluates to:\r\n  \u003e\u003e S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\r\n  S =\r\n      5.5833\r\nPlease round-off  to the nearest whole number. Therefore, the final output for the above example is . ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 435px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 757.578125px 217.5px; transform-origin: 757.578125px 217.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eGiven two positive integers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we define the function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"46\" height=\"19\" style=\"width: 46px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the sum of the reciprocals of the products of all combinations of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003e1\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e taken \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e at a time. In Matlab we say that:    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eR = sum(1./prod(nchoosek(a,b)'))\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. For example if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36.5\" height=\"18\" style=\"width: 36.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 754.578125px 30px; transform-origin: 754.578125px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; R = sum(1./prod(nchoosek(1:5,3)'))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  R =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e      0.7083\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven a positive integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, we are asked to evaluate the following double summation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: left; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg 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\" width=\"395.5\" height=\"46\" style=\"width: 395.5px; height: 46px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIn which, in Matlab, evaluates to:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 754.578125px 30px; transform-origin: 754.578125px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  \u0026gt;\u0026gt; S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e  S =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 754.578125px 10px; transform-origin: 754.578125px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e      5.5833\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePlease round-off \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eS\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to the nearest whole number. Therefore, the final output for the above example is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37.5\" height=\"18\" style=\"width: 37.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = S(x)\r\n    s = round(sum(arrayfun(@(j) sum(arrayfun(@(i) sum(1./prod(nchoosek(1:j,i)')),2:j-1)),3:x)));\r\nend","test_suite":"%%\r\nx = 1:10;\r\ns_correct = [0 0 1 3 6 9 14 19 25 32];\r\nassert(isequal(arrayfun(@S,x),s_correct))\r\n%%\r\nx = 15;\r\ns_correct = 81;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 20;\r\ns_correct = 154;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 25:25:500;\r\ns_correct = [250 1095 2552 4625 7318 10630 14563 19118 24294 30093 36514 43558 51225 59515 68428 77965 88125 98908 110315 122346];\r\nassert(isequal(arrayfun(@S,x),s_correct))\r\n%%\r\nx = 1234;\r\ns_correct = 753724;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 12345;\r\ns_correct = 76094591;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 123456;\r\ns_correct = 7619358524;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1234567;\r\ns_correct = 762061661646;\r\nassert(isequal(S(x),s_correct))\r\n%%\r\nx = 1001001:1002001;\r\ns = arrayfun(@S,x);\r\nss = floor([mean(s) median(s) mode(s) std(s)])\r\nss_correct = [501489254654 501489212905 500988594352 289538048];\r\ns = s(floor(logspace(0,log10(length(x)),10)));\r\ns_correct = [500988594352 500989595339 500991597317 500997603276 501008614294 501033639784 501087696976 501202828490 501452158566 501990081457];\r\nassert(isequal(s,s_correct))\r\nassert(isequal(ss,ss_correct))\r\n%%\r\nfiletext = fileread('S.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'java') || contains(filetext, 'py') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":255988,"edited_by":255988,"edited_at":"2023-01-07T06:32:38.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2023-01-05T10:59:18.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-05T01:20:27.000Z","updated_at":"2026-01-22T20:49:35.000Z","published_at":"2023-01-05T10:36:53.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven two positive integers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\\\\ge b\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we define the function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eR(a,b)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the sum of the reciprocals of the products of all combinations of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e taken \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at a time. In Matlab we say that:    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eR = sum(1./prod(nchoosek(a,b)'))\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. For example if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea=5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb=3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e R = sum(1./prod(nchoosek(1:5,3)'))\\n  R =\\n      0.7083]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven a positive integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, we are asked to evaluate the following double summation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(x) = \\\\sum_{n=3}^{x} \\\\sum_{k=2}^{n-1} R(n,k).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex=5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS(5) = \\\\sum_{n=3}^{5} \\\\sum_{k=2}^{n-1} R(n,k) = \\\\sum_{k=2}^2 R(3,k)+\\\\sum_{k=2}^3 R(4,k)+\\\\sum_{k=2}^4 R(5,k).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn which, in Matlab, evaluates to:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[  \u003e\u003e S = sum(arrayfun(@(i) sum(1./prod(nchoosek(1:3,i)')),2:2))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:4,i)')),2:3))+sum(arrayfun(@(i) sum(1./prod(nchoosek(1:5,i)')),2:4))\\n  S =\\n      5.5833]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease round-off \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to the nearest whole number. Therefore, the final output for the above example is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"summation\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"summation\"","current_player":null,"sort":"map(difficulty_value,0,0,999) 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