We have \begin{equation} \begin{split} \sum_{n = 0}^N n & = 0 + 1 + 2 + \cdots + (N - 2) + (N - 1) + N \\ & = N + (N - 1) + (N - 2) + \cdots + 2 + 1 + 0 \\ & = \sum_{n = 0}^N (N - n) \end{split} \end{equation} $\square$