// Numbas version: exam_results_page_options {"name": "Binomial series for Natural exponent", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Binomial series for Natural exponent", "ungrouped_variables": ["a", "b", "n", "answer", "k", "negb"], "preamble": {"css": "", "js": ""}, "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "

Given the expression \$$(\\var{a}-\\var{negb}x)^{\\var{n}}\$$

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By using the binomial series expansion, calculate the coefficient of \$$x^{\\var{k}}\$$.

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The coefficient of \$$x^{\\var{k}}\$$ is  [[0]]

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

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\$$(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}\$$

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

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So the coefficient of \$$x^{\\var{k}}\$$ is given by \$$\\tbinom{\\var{n}}{\\var{k}} \\times \\var{a}^{\\var{n}-\\var{k}} \\times (\\var{b})^{\\var{k}}=\\var{c}\$$

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Binomial series for Natural exponent

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