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If $a_{r}$ is the coefficient $x^{r}$ in the expansion of ${(1 + x + x^{2})}^{n}$ , then $a_{1} - 2 a_{2} + 3 a_{3} - \dots - 2 n a_{2 n} =$

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n

2 n

answer is C.

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Detailed Solution

{(1 + x + x^{2})}^{n} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots + a_{2 n} x^{2 n}

Differentiating on both sides with respect to $x$ , we get

$n . {(1 + x + x^{2})}^{n - 1} (1 + 2 x) = a_{1} + 2 a_{2} x + 3 a_{3} x^{2} + \dots + 2 n . a_{2 n} x^{2 n - 1}$

$put x = - 1$

$n (- 1) = a_{1} - 2 a_{2} + 3 a_{3} + \dots - 2 n a_{2 n}$

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