Question

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

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

Answer 1

Answer: (A)

$510$

$2\times 2^2+3\times 2^3+4\times 2^4...n\times 2^n=2^{n+10}$

It is Arithmetic-geometric progression

$2,3,4...AP$

$2^3,2^3,2^4....GP$

let

$S=2\times 2^2+{3.2}^3+{4.2}^4...n{.2}^n$

$2S=$ ${2.2}^3+{3.2}^4...(n-1).2^n+n.2^{n+1}$

__________________________________________

$-S={2.2}^2+2^3+2^4...+2^n-n.2^{n+1}$

$-S={2.2}^2+2^3\left(\frac{2^{n-3}-1}{2-1}\right)-n.2^{n+1}$

$-S={2.2}^2+2^3(2^{n-3}-1)-n{.2}^{n+1}$

$=2^3+2^n-2^3-n{.2}^{n+1}$

$=2^n-2n{.2}^n$

$S=2^n(2n-1)$

$2^n.2^{10}=2^n(2n-1)$

$2^{10}=2n-1$

$1024=2n-1$

$1025=2n$

$n=\frac{1025}{2}$

$n=512.5$

integer $=512$