If $a_{r}$ is the coefficient of $x^{r}$ , in the expansion of ${(1 + x + x^{2})}^{n}$, then $a_{1} - 2a_{2} + 3a_{3} - ... - 2n{\ a}_{2n}$ is equal to

Question

Dr. Abubacker · Accepted Answer

(c)
Let us take
$a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{2n}x^{2n} = {(1 + x + x^{2})}^{n}$
On differentiating both sides w.r.t. $x$, we get
$a_{1} + {2a}_{2}x + ... + {2na}_{2n}x^{2n - 1} = n\left( 1 + x + x^{2} \right)^{n - 1}(2x + 1)$
Put $x = - 1$
$\Rightarrow \ a_{1} - {2a}_{2} + {3a}_{3} - ... - {2na}_{2n} = - n$

Q. If $a_{r}$ is the coefficient of $x^{r}$ , in the expansion of ${(1 + x + x^{2})}^{n}$, then $a_{1} - 2a_{2} + 3a_{3} - ... - 2n{\ a}_{2n}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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