If a_k is the coefficient of x^k in the expan | vedclass

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(D) We have,$(1+x+x^2)^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_{2n}x^{2n}$.On differentiating both sides with respect to $x$,we get:$n(1+x+x^2)^

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$n \cdot 3^n$

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(D) We have,$(1+x+x^2)^n = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_{2n}x^{2n}$.On differentiating both sides with respect to $x$,we get:$n(1+x+x^2)^{n-1}(1+2x) = a_1 + 2a_2x + 3a_3x^2 + \ldots + 2na_{2n}x^{2n-1}$.Now,putting $x=1$,we get:$n(1+1+1)^{n-1}(1+2(1)) = a_1 + 2a_2 + 3a_3 + \ldots + 2na_{2n}$.$n(3)^{n-1}(3) = a_1 + 2a_2 + 3a_3 + \ldots + 2na_{2n}$.Therefore,$a_1 + 2a_2 + 3a_3 + \ldots + 2na_{2n} = n \cdot 3^n$.

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n$ for $k=0, 1, 2, \ldots, 2n$,then $a_1+2a_2+3a_3+\ldots+2na_{2n}$ is equal to

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