Textbook Problem

Use Pascal’s formula to prove by mathematical induction that if n is an integer and $n\ge 1$ , then

$\begin{array}{c}{\displaystyle \underset{i=2}{\overset{n+1}{\sum }}\left({}_2^i\right)}=\left({}_2^2\right)+\left({}_2^3\right)+\mathrm{...}+\left({}_2^{n+1}\right)\\ \left({}_3^{n+2}\right)\end{array}$