Question

The positive integer $n$ for which $2.2^2+3.2^3+4.2^4+\cdots +n.2^n=2^{n+10}$ is

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

Answer 1

Answer: (D)

$513$

$2^{n+10}=2\cdot 2^2+3\cdot 2^3+4\cdot 2^4+\cdots +n\cdot 2^n\to \left(1\right)$

Multiply $\left(1\right)$ by $2$
$2^{n+11}=2\cdot 2^3+3\cdot 2^4+4\cdot 2^5+\cdots +n\cdot 2^{n+1}\to \left(2\right)$

Subtracting $\left(2\right)$ from $\left(1\right)$ by shifting one place, we get
$-2^{n+10}=2\cdot 2^2+2^3+2^4+\cdots +2^n-n\cdot 2^{n+1}\to \left(3\right)$

$n\cdot 2^{n+1}-2^{n+10}=2+2^1+2^2+\cdots +2^n$

$\Rightarrow n\cdot 2^n-2^9\cdot 2^n=1+\frac{2^n-1}{2-1}$

$\Rightarrow n-1=2^9$

$n=513$