// Numbas version: exam_results_page_options {"name": "First 3 terms of Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"k": {"description": "", "templateType": "anything", "name": "k", "group": "Ungrouped variables", "definition": "random(2..{n}-1)"}, "a": {"description": "", "templateType": "randrange", "name": "a", "group": "Ungrouped variables", "definition": "random(2..10#1)"}, "n": {"description": "", "templateType": "randrange", "name": "n", "group": "Ungrouped variables", "definition": "random(4..9#1)"}, "c": {"description": "", "templateType": "anything", "name": "c", "group": "Ungrouped variables", "definition": "comb({n},{k})*{a}^({n}-{k})*{b}^{k}"}, "b": {"description": "", "templateType": "randrange", "name": "b", "group": "Ungrouped variables", "definition": "random(2..12#1)"}}, "variable_groups": [], "name": "First 3 terms of Binomial series for Natural exponent", "preamble": {"css": "", "js": ""}, "statement": "

Given the expression \\((\\var{a}+\\var{b}x)^{\\var{n}}\\)

", "metadata": {"description": "

Find the first 3 terms of Binomial series having a Natural exponent

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "advice": "

The binomial series expansion for an expression of the form \\((a+bx)^n\\) where \\(n\\) is a Natural number is given by:

\n

\\((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}\\)

\n

In this example  \\(n=\\var{n}\\),  \\(k=\\var{k}\\),  \\(a=\\var{a}\\)  and  \\(b=\\var{b}\\).

\n

So the first three terms of the binomial series expansion are:

\n

\\(\\var{a}^{\\var{n}}+\\tbinom{\\var{n}}{\\var{1}}*\\var{a}^{\\var{n}-1}*\\var{b}^{1}+\\tbinom{\\var{n}}{2}*\\var{a}^{\\var{n}-2}*\\var{b}^{2}\\)

\n

\\(=\\simplify{{a}^{n}}+\\simplify{{n}*{a}^({n}-1)*{b}}x+\\simplify{{n}*{n-1}*{a}^{{n}-2}*{b}^2/2x^2}\\)

\n

", "parts": [{"showFeedbackIcon": true, "marks": "3", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "prompt": "

Write out the first three terms of the binomial expansion.

", "scripts": {}, "answer": "{a}^{n}+{n}*{a}^({n}-1)*{b}x+{n}*{n-1}*{a}^({n}-2)*{b}^2/2x^2", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff", "expectedvariablenames": [], "type": "jme", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showpreview": true}], "ungrouped_variables": ["a", "b", "n", "c", "k"], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "tags": [], "rulesets": {}, "functions": {}, "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/"}]}