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

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

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

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

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

answer is A.

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Detailed Solution

$\sum_{k = 0}^{2005} (a + kd) \cdot C_{k} = a \sum_{k = 0}^{2005} C_{k} + d \sum_{k = 0}^{2005} k \cdot C_{k} = a \cdot 2^{2005} + d \cdot 2005 \cdot 2^{2 (104} = (2 a + 2005 d) \cdot 2^{2004}$

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