// Numbas version: exam_results_page_options
{"name": "Simon's copy of First 3 terms of Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Simon's copy of First 3 terms of Binomial series for Natural exponent", "statement": "<p>Given the expression \\((\\var{a}+\\var{b}x)^{\\var{n}}\\)</p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}", "name": "c", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"definition": "random(2..7#1)", "name": "a", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "b": {"definition": "random(2..8#1)", "name": "b", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "n": {"definition": "random(4..9#1)", "name": "n", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "k": {"definition": "random(2..{n}-1)", "name": "k", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "rulesets": {}, "ungrouped_variables": ["a", "b", "n", "c", "k"], "preamble": {"css": "", "js": ""}, "tags": [], "functions": {}, "advice": "<p>The binomial series expansion for an expression of the form \\((a+bx)^n\\) where \\(n\\) is a Natural number is given by:</p>\n<p style=\"padding-left: 30px;\">\\((a+bx)^n=\\tbinom{n}{0}(a)^n(bx)^{0}+\\tbinom{n}{1}(a)^{n-1}(bx)^{1}+\\tbinom{n}{2}(a)^{n-2}(bx)^{2}+...\\tbinom{n}{k}(a)^{n-k}(bx)^{k}+...\\tbinom{n}{n}(a)^{0}(bx)^{n}\\)</p>\n<p>In this example &nbsp;\\(n=\\var{n}\\),&nbsp; \\(a=\\var{a}\\) &nbsp;and &nbsp;\\(b=\\var{b}\\).</p>\n<p>So the first three terms of the binomial series expansion are:</p>\n<p style=\"padding-left: 30px;\">&nbsp;\\(\\var{a}^{\\var{n}}+\\tbinom{\\var{n}}{\\var{1}}\\times\\var{a}^{\\var{n}-1}\\times\\var{b}^{1}+\\tbinom{\\var{n}}{2}\\times\\var{a}^{\\var{n}-2}\\var{b}^{2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{a}^{n}}+\\simplify{{n}*{a}^({n}-1)*{b}}x+\\simplify{{n}*{n-1}*{a}^{{n}-2}*{b}^2/2x^2}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "parts": [{"checkingAccuracy": 0.001, "showFeedbackIcon": true, "vsetRange": [0, 1], "expectedVariableNames": [], "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "answer": "{a}^{n}+{n}*{a}^({n}-1)*{b}x+{n}*{n-1}*{a}^({n}-2)*{b}^2/2x^2", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "prompt": "<p>Write out the first three terms of the binomial expansion.&nbsp;&nbsp;</p>", "answerSimplification": "all, canonicalOrder, !noLeadingMinus", "vsetRangePoints": 5, "showPreview": true, "failureRate": 1, "showCorrectAnswer": true, "checkVariableNames": false, "unitTests": [], "marks": "3", "type": "jme", "variableReplacements": [], "scripts": {}}], "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Find the first 3 terms of Binomial series having a&nbsp;Natural exponent</p>"}, "extensions": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}