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

  • question_answer
    If \[|x|<1\], then in the expansion of \[{{(1+2x+3{{x}^{2}}+4{{x}^{3}}+....)}^{1/2}},\] the coefficient of \[{{x}^{n}}\]is

    A) n

    B) \[n+1\]

    C) 1

    D) - 1

    Correct Answer: C

    Solution :

    Since  \[1+2x+3{{x}^{2}}+4{{x}^{3}}+....\infty ={{(1-x)}^{-2}}\] Therefore, we have \[{{(1+2x+3{{x}^{2}}+4{{x}^{3}}+....\infty )}^{1/2}}={{\{{{(1-x)}^{-2}}\}}^{1/2}}\] \[={{(1-x)}^{-1}}=1+x+{{x}^{2}}+....+{{x}^{n}}+....\infty \] \ The coefficient of \[{{x}^{n}}=1\].


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