Question

If $(2\times 2^2)+(3\times 2^3)+(4\times 2^4)...+(n\times 2^n)=2^{n+10}$, then $n=$

• A. $510$
• B. $512$
• C. $513$
• D. $508$

Answer 1

Answer: (C)

$513$

Let $S=2.2^2+3.2^3+4.2^4+...n.2^n$ ...(1)
Multiply this by $2$, we get
$2S=2.2^3+3.2^4+4.2^5+...n.2^{n+1}$ ...(2)
$\left(1\right)-\left(2\right)$, we get
$-S=2.2^2+2^3+2^4+...+2^n-n{2}^{n+1}$
$\Rightarrow -S={2.2}^2+2^3\left(\frac{2^{n-2}-1}{2-1}\right)-n{2}^{n+1}$
$\Rightarrow -S={2.2}^3+2^{n+1}-2^3-{n2}^{n+1}\Rightarrow -S=-(n-1)2^{n+1},S=(n-1)2^{n+1}\therefore 2^{n+10}=(n-1)2^{n+1}\Rightarrow 2^9=(n-1)\Rightarrow n=513$