The positive integer $n$ for which $2 imes 2^2+3 imes 2^3+4 imes 2^4+\ldots+n imes 2^n=2^{n+

Question

The positive integer $n$ for which $2 	imes 2^2+3 	imes 2^3+4 	imes 2^4+\ldots+n 	imes 2^n=2^{n+6}$, is

Professor PC Thomas Classes · Answer

Arithmetico-geometric series. $S_n = \sum_{k=2}^n k 2^k = (n-1)2^{n+1} + 4$. Equating to given form yields $n$.