Closed Form for Triangular Numbers/Proof by Polygonal Numbers

Proof
Triangular numbers are $k$-gonal numbers where $k = 3$.

From Closed Form for Polygonal Numbers we have that:
 * $P \left({k, n}\right) = \dfrac n 2 \left({\left({k - 2}\right) n - k + 4}\right)$

Hence:

Hence the result.