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Standard XI

Mathematics

Binomial Coefficients

If Ck is the ...

Question

If $C_{k}$ is the coefficient of in the expansion of ${(1 + x)}^{2005}$ and if a, d are real numbers then $\sum_{k = 0}^{2005} (a + k d) . C_{k} =$

(2 a + 2005 d) 2^{2004}

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(2 a + 2005 d) 2^{2005}

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(2 a + 2004 d) 2^{2005}

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(2 a + 2004 d) 2^{2005}

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Solution

The correct option is A $(2 a + 2005 d) 2^{2004}$
$\sum_{k = 0}^{2005} (a + k d) . C_{k} = a \sum_{k = 0}^{2005} C_{k} + d \sum_{k = 0}^{2005} k . C_{k} = a {.2}^{2005} + d .2005 {.2}^{2004} = (2 a + 2005 d) {.2}^{2004}$

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If Ck is the coefficient of in the expansion of (1+x)2005 and if a, d are real numbers then ∑2005k=0(a+kd).Ck=

The correct option is A (2a+2005 d)22004∑2005k=0(a+kd).Ck=a∑2005k=0Ck+d∑2005k=0k.Ck=a.22005+d.2005.22004=(2a+2005d).22004

If $C_{k}$ is the coefficient of in the expansion of ${(1 + x)}^{2005}$ and if a, d are real numbers then $\sum_{k = 0}^{2005} (a + k d) . C_{k} =$

The correct option is A $(2 a + 2005 d) 2^{2004}$
$\sum_{k = 0}^{2005} (a + k d) . C_{k} = a \sum_{k = 0}^{2005} C_{k} + d \sum_{k = 0}^{2005} k . C_{k} = a {.2}^{2005} + d .2005 {.2}^{2004} = (2 a + 2005 d) {.2}^{2004}$