Question

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)$ for $k=0,1,2,\dots 2n$ then the value of $\sum _{k=1}^{2n}k{a}_k$ is

• A. $–a_0$
• B. $3^n$
• C. $n.3^{n+1}$
• D. $n.3^n$

Answer 1

The correct option is D

$n.3^n$

Explanation for the correct option:

Step 1. Write the expansion of ${(1+x+x^2)}^n$:

$(1+x+x^2)=a_0+a_{1}x+a_2{x}^2+.....+a_k{x}^k+.....+a_{2n}{x}^{2n}$

Differentiate the above equation with respect to $x$,

$n{(1+x+x^2)}^{n-1}\left(2x+1\right)=a_1+2a_{2}x+.....+k{a}_k{x}^{k-1}+.....+2n{a}_{2n}{x}^{2n-1}$

Step 2. Put $x=1$ in above equation, we get

$\left(n\right)3^{n-1}\left(3\right)=a_1+2a_2+.....+k{a}_k+.....+2n{a}_{2n}$

$\Rightarrow$ $\left(n\right)3^n=a_1+2a_2+.....+k{a}_k+.....+2n{a}_{2n}$

$\therefore$$\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{1}}^{\mathbf{2}\mathbf{n}}\mathit{k}{\mathit{a}}_{\mathbf{k}}\mathbf{}\mathbf{=}\mathbf{}\mathit{n}\mathbf{}\mathbf{.}\mathbf{}{\mathbf{3}}^{\mathbf{n}}$

Hence, Option ‘D’ is Correct.