Textbook Problem

For every integer $n\ge 1$ ,
$\begin{array}{l}\left({}_0^n\right)-\frac{1}{2}\left({}_1^n\right)+\frac{1}{2^2}\left({}_2^n\right)-\frac{1}{2^3}\left({}_3^n\right)\\ +\mathrm{...}+{\left(-1\right)}^{n-1}\frac{1}{2^{n-1}}\left({}_{n-1}^n\right)=\{{}_{\frac{1}{2^{n-1}}\text{if }n\text{in}\text{odd}\text{.}}^{0\text{if}n\text{is}\text{eveen}}\end{array}$