If ak is the coefficient of xk in the expansion of 1-x-x2n for k-0-1-2- - 2n- then a1-2a2-3a3- -2na2n-

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Index of (r+1)th Term from End When Counted from Beginning

If ak is th...

Question

If $a_{k}$ is the coefficient of $x^{k}$ in the expansion of $(1 + x + x^{2})^{n}$ for $k = 0, 1, 2,$ $\dots$ , $2 n$ , then $a_{1} + 2 a_{2} + 3 a_{3} + \dots + 2 n a_{2_{n}} =$

- a_{0}

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3^{n}

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n {.3}^{n}

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n - 3^{n}

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(1 + X + X^{2})^{n} = \sum_{r = 0}^{2 n} a_{r} x^{r}

a_{1} - 2 a_{2} + 3 a_{3} . . . . - 2 n a_{2 n}

(1 + x + x^{2})^{n} = 2 n \sum r = 0 a_{r} x^{r}

a_{1} - 2 a_{2} + 3 a_{3} - \dots . - 2 n a_{2 n} = \dots

a_{0}, a_{1}, a_{2}, . . . . .

(1 + x + x^{2})^{n}

(a_{0} + a_{3} + a_{6} + . . .) = (a_{1} + a_{4} + a_{7} + . . . .) = = (a_{2} + a_{5} + a_{8} + . . .) = 3^{n - 1}

x^{r} [0 \leq r \leq (n - 1)]

{(x + 3)}^{n - 1} + {(x + 3)}^{n - 2} (x + 2) + {(x + 3)}^{n - 3} {(x + 2)}^{2} + . . . . . + {(x + 2)}^{n - 1}

x

{(x - \frac{1}{x^{2}})}^{3 n}

n

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