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JCECE Engineering JCECE Engineering Solved Paper-2011

  • question_answer
    If\[S=\sum\limits_{n=2}^{\infty }{^{n}{{C}_{2}}}\frac{{{3}^{n-2}}}{n!}\], then\[2s\]equals to

    A) \[{{e}^{3/2}}\]                                  

    B) \[{{e}^{3}}\]

    C) \[{{e}^{-3/2}}\]                                

    D) \[{{e}^{-3}}\]

    Correct Answer: B

    Solution :

    We have,\[S=\sum\limits_{n=2}^{\infty }{^{n}{{C}_{2}}}\frac{{{3}^{n-2}}}{n!}\]                 \[=\sum\limits_{n=2}^{\infty }{\frac{n!}{(n-2)!2n!}\cdot {{3}^{n-2}}}\]                 \[=\frac{1}{2}\sum\limits_{n=2}^{\infty }{\frac{{{3}^{n-2}}}{(n-2)!}=\frac{1}{2}{{e}^{3}}}\] \[\Rightarrow \]               \[2S={{e}^{3}}\]


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