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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Self Evaluation Test - Binomial Theorem

  • question_answer
    The coefficient of \[{{x}^{n}}\] in the polynomial\[(x+{{\,}^{n}}{{C}_{0}})(x+3.{{\,}^{n}}{{C}_{1}})(x+5.{{\,}^{n}}{{C}_{2}})...(x+{{(2n+1)}^{n}}{{C}_{n}})\] is

    A) \[n{{.2}^{n}}\]

    B) \[~n{{.2}^{n+1}}\]

    C) \[(n+1){{.2}^{n}}\]

    D) \[n{{.2}^{n}}+1\]

    Correct Answer: C

    Solution :

    [c] \[(x+{{\,}^{n}}{{C}_{0}})(x+3.{{\,}^{n}}{{C}_{1}})(x+5.{{\,}^{n}}{{C}_{2}})....\]
    \[(x+(2n+1).{{\,}^{n}}{{C}_{n}})\]
    \[={{x}^{n+1}}+{{x}^{n}}\{{{\,}^{n}}{{C}_{0}}+3.{{\,}^{n}}{{C}_{1}}+5.{{\,}^{n}}{{C}_{2}}+....\]
    \[+(2n+1).{{\,}^{n}}{{C}_{n}}\}+......\]
    Coeff. of \[{{x}^{n}}\]
    \[={{\,}^{n}}{{C}_{0}}+3.{{\,}^{n}}{{C}_{1}}+5.{{\,}^{n}}{{C}_{2}}+....+(2n+1).{{\,}^{n}}{{C}_{n}}\]
    \[=1+({{\,}^{n}}{{C}_{1}}+2.{{\,}^{n}}{{C}_{1}})+({{\,}^{n}}{{C}_{2}}+4.{{\,}^{n}}{{C}_{2}})+...\]
                                        \[+{{(}^{n}}{{C}_{n}}+2n.{{\,}^{n}}{{C}_{n}})\]
    \[=(1+{{\,}^{n}}{{C}_{1}}+...+{{\,}^{n}}{{C}_{n}})+2({{\,}^{n}}{{C}_{1}}+2{{\,}^{n}}{{C}_{2}}+...+n.{{\,}^{n}}{{C}_{n}})\]
                \[={{2}^{n}}+2\left[ n+2.\frac{n(n-1)}{2!}+3.\frac{n(n-1)(n-2)}{3!}+...+n.1 \right]\]\[={{2}^{n}}+2n[1+{{\,}^{n-1}}{{C}_{1}}+{{\,}^{n-1}}{{C}_{2}}+....+{{\,}^{n-1}}{{C}_{n-1}}]\]
    \[={{2}^{n}}+2n{{.2}^{n-1}}={{2}^{n}}(1+n)=(n+1){{.2}^{n}}\]


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