// 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}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["b", "n", "T1", "T2", "T3"], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "

Find the first three terms in the binomial series expansion having a non-Natural exponent

"}, "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": "

Input the first three tems in the binomial series expansion.

", "variableReplacements": [], "scripts": {}, "marks": "3", "expectedvariablenames": []}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The binomial series expansion for an expression of the form \$$(1+bx)^{n}\$$ where \$$n\$$ is a non-natural number, is given by:

\n

\$$(1+bx)^{n}=1+\\tbinom{n}{1}(bx)^{1}+\\tbinom{{n}}{2}(bx)^{2}+\\tbinom{{n}}{3}(bx)^{3}.......\$$

\n

In this example  \$$power=\\frac{1}{\\var{n}}\$$,   and  \$$b=\\var{b}\$$.

\n

So the first three terms are

\n

\$$(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}\$$

\n

\$$(1+\\var{b}x)^{\\frac{1}{\\var{n}}}=\\var{T1}+\\var{T2}x+\\var{T3}x^2\$$

", "variable_groups": [], "statement": "

Given the expression \$$(1+\\var{b}x)^{\\frac{1}{\\var{n}}}\$$

", "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/"}]}