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

  • question_answer
    If \[{{C}_{0}},{{C}_{1}},\,{{C}_{2}}{{,}^{.}}.......,\,\,{{C}_{15}}\] are binomial coefficients in \[{{(1+x)}^{15}}\], then\[\frac{{{C}_{1}}}{{{C}_{0}}}+2\frac{{{C}_{2}}}{{{C}_{1}}}+3\frac{{{C}_{3}}}{{{C}_{2}}}+....+15\frac{{{C}_{15}}}{{{C}_{14}}}=\]

    A) 60

    B) 120

    C) 64

    D) 124

    Correct Answer: B

    Solution :

    [b] General term of the given series is \[r\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=n+1-r\] By taking summation over n, we get \[\sum\limits_{1}^{15}{r\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=\sum\limits_{n=1}^{15}{(n+1-r)=\sum\limits_{1}^{15}{(16-r)}}}\] \[=16\times 15-\frac{1}{2}\cdot 15\times 16\] By using sum of n natural numbers \[=\frac{n(n+1)}{2}\] \[=240-120=120\]


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