Question

The positive integer $n$ for which $2\times 2^2+3\times 2^3+4\times 2^4+\dots +n\times 2^n=2^{n+10}$ is

• A. $510$
• B. $511$
• C. $512$
• D. $513$

Answer 1

The correct option is D $513$

We have,
$2^{n+10}=2\times 2^2+3\times 2^3+4\times 2^4+\dots +n\times 2^n$
$\Rightarrow 2\left(2^{n+10}\right)=2\times 2^3+3\times 2^4+\dots +(n-1)\times 2^n+n\times 2^{n+1}$
Subtracting , we get
$-2^{n+10}=2\times 2^2+2^3+2^4+\dots +2^n-n\times 2^{n+1}$
$\Rightarrow -2^{n+10}=8+\frac{8(2^{n-2}-1)}{2-1}-n{.2}^{n+1}$
$\Rightarrow -2^{n+10}=8+2^{n+1}-8-n\times 2^{n+1}$
$\Rightarrow -2^{n+10}=2^{n+1}-\left(n\right)2^{n+1}$
$\Rightarrow 2^{n+10}=\left(n\right)2^{n+1}-2^{n+1}$
$\Rightarrow 2^9=n-1\Rightarrow n=513$