For arbitrary positive integral $n,$ find the sum $$2^{n}+2^{n-1}+2^{n-2}+\ldots+2^{-n+1}+2^{-n}.$$
My solution: $$\begin{align} 2^{n}+2^{n-1}+2^{n-2}+\ldots+2^{-n+1}+2^{-n}&=2^{n}[2^{0}+2^{-1}+2^{-2}+2^{-3}+\dots 2^{-n}]\\ &=2^{n}\sum_{0}^{n}2^{-n}\\ &=2^{n}[2-\frac{1}{2^{2n}}]\\ &=2^{n+1}-2^{-n}. \end{align}$$ Is my solution correct?