\begin{align*} \frac{k(k+1)}{2} + k+1 \amp = \frac{k(k+1)}{2} + \frac{2(k+1)}{2}\\ \amp = \frac{k(k+1) + 2(k+1)}{2}\\ \amp = \frac{(k+2)(k+1)}{2} \end{align*}