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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Binomial theorem for any index

  • question_answer
    The coefficient of \[{{x}^{n}}\] in the expansion of \[{{(1+x+{{x}^{2}}+....)}^{-n}}\] is

    A) 1

    B) \[{{(-1)}^{n}}\]

    C) n

    D) \[n+1\]

    Correct Answer: B

    Solution :

    We have,  \[{{(1+x+{{x}^{2}}+...)}^{-n}}={{[{{(1-x)}^{-1}}]}^{-n}}={{(1-x)}^{n}}\] \[={{\,}^{n}}{{C}_{0}}-{{\,}^{n}}{{C}_{1}}x+{{\,}^{n}}{{C}_{2}}{{x}^{2}}+...+{{(-1)}^{n}}\,{{\,}^{n}}{{C}_{n}}.{{x}^{n}}\] Coefficient of\[{{x}^{n}}\]is\[{{(-1)}^{n}}{{\,}^{n}}{{C}_{n}}={{(-1)}^{n}}\].


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