// Numbas version: exam_results_page_options
{"name": " Binomial series for non-Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["b", "n", "T1", "T2", "T3"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Find the first three terms in the binomial series expansion having a non-Natural exponent</p>"}, "tags": [], "name": " Binomial series for non-Natural exponent", "extensions": [], "functions": {}, "parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "checkvariablenames": false, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "answer": "{T1}+{T2}x+{T3}x^2", "prompt": "<p>Input the first three tems in&nbsp;the binomial series expansion.&nbsp;</p>", "variableReplacements": [], "scripts": {}, "marks": "3", "expectedvariablenames": []}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The binomial series expansion for an expression of the form \\((1+bx)^{n}\\) where \\(n\\) is a non-natural number, is given by:</p>\n<p style=\"padding-left: 30px;\">\\((1+bx)^{n}=1+\\tbinom{n}{1}(bx)^{1}+\\tbinom{{n}}{2}(bx)^{2}+\\tbinom{{n}}{3}(bx)^{3}.......\\)</p>\n<p>In this example &nbsp;\\(power=\\frac{1}{\\var{n}}\\), &nbsp; and &nbsp;\\(b=\\var{b}\\).</p>\n<p>So the&nbsp;first three terms are</p>\n<p style=\"padding-left: 30px;\">\\((1+\\var{b}x)^{\\frac{1}{\\var{n}}}=1+\\tbinom{\\frac{1}{\\var{n}}}{1}(\\var{b}x)^{1}+\\tbinom{\\frac{1}{\\var{n}}}{2}(\\var{b}x)^{2}\\)</p>\n<p style=\"padding-left: 30px;\">\\((1+\\var{b}x)^{\\frac{1}{\\var{n}}}=\\var{T1}+\\var{T2}x+\\var{T3}x^2\\)</p>", "variable_groups": [], "statement": "<p>Given the expression \\((1+\\var{b}x)^{\\frac{1}{\\var{n}}}\\)</p>", "variables": {"n": {"name": "n", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(2..6#1)"}, "T2": {"name": "T2", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "{b}/{n}"}, "b": {"name": "b", "group": "Ungrouped variables", "templateType": "randrange", "description": "", "definition": "random(2..12#1)"}, "T1": {"name": "T1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "1"}, "T3": {"name": "T3", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "{b}^2*(1/{n}*(1/{n}-1))/2"}}, "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}