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

  • question_answer
    If \[{{a}_{n}}=2n+1\] and \[{{C}_{r}}={{\,}^{n}}{{C}_{r}}\] then\[{{a}_{0}}C_{0}^{2}+{{a}_{1}}C_{1}^{2}+{{a}_{2}}C_{2}^{2}+........{{a}_{n}}C_{n}^{2}=\]

    A) \[(n-1){{(}^{2n}}{{C}_{n}})\]

    B) \[n{{(}^{2n}}{{C}_{n}})\]

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

    D) \[(n+1){{(}^{n}}{{C}_{n/2}})\]  

    Correct Answer: C

    Solution :

    [c] \[{{S}_{n}}={{a}_{0}}{{C}_{0}}^{2}+{{a}_{1}}{{C}_{1}}^{2}+{{a}_{2}}{{C}_{2}}^{2}+.......+{{a}_{n}}{{C}_{n}}^{2}\] \[\frac{{{S}_{n}}={{a}_{n}}{{C}_{n}}^{2}+{{a}_{n-1}}{{C}_{n-1}}^{2}+{{a}_{n-2}}^{2}+....+{{a}_{0}}{{C}_{0}}^{2}}{2{{S}_{n}}=({{a}_{0}}+{{a}_{n}}){{C}_{0}}^{2}+({{a}_{1}}+{{a}_{n-1}}){{C}_{1}}^{2}+...+({{a}_{n}}+{{a}_{0}}){{C}_{n}}^{1}}\]\[=(2n+2)({{C}_{0}}^{2}+{{C}_{1}}^{2}+{{C}_{2}}^{2}+.....+{{C}_{n}}^{2})\] \[\therefore \,\,\,\,{{S}_{n}}={{(n+1)}^{2n}}{{C}_{n}}\]             \[[\because \,\,\,\,\,{{a}_{0}}+{{a}_{n}}={{a}_{1}}+{{a}_{n-1}}+.......=2n+2]\]


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