Difference Between Adjacent Polygonal Numbers is Triangular Number

Theorem
Let $P \left({k, n}\right)$ be the $n$th $k$-gonal number.

Then:
 * $P \left({k + 1, n}\right) - P \left({k, n}\right) = T_{n - 1}$

where $T_n$ is the $n$th triangular number.

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

Thus: