Elementary problems Find $n$ such that $n$ is a positive integer satisfying the following equation.
$$2(2^2)+3(2^3)+4(2^4)+\ldots+n(2^n)=2^{n+10}$$
Can anybody help ? I can't believe this is an elementary school problem...

Let $\displaystyle S=2\cdot2^2+3\cdot2^3+\cdots+(n-1)2^{n-1}+n\cdot2^n$
$$\implies2S=2\cdot2^3+3\cdot2^4+\cdots+(n-1)2^{n}+n\cdot2^{n+1}$$
$$S-2S=2\cdot2^2+(3-2)2^3+(4-3)2^4+\{n-(n-1)\}2^n-n\cdot2^{n+1}$$
$$\implies -S=2\cdot2^2-n\cdot2^{n+1}+\underbrace{(2^3+2^4+\cdots+2^n)}_{\text{Geomteric Series}}$$
$$\implies-S=2\cdot2^2-n\cdot2^{n+1}+2^3\cdot\frac{(2^{n-2}-1)}{(2-1)}$$
$$\implies S=(n-1)2^{n+1}$$ which needs to be $\displaystyle2^{n+10}\implies n-1=2^9$
Reference : Arithmetico-geometric sequence
