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
We have from Closed Form for Polygonal Numbers that:
 * $P \left({k, n}\right) = \dfrac {n \left({2 + \left({n - 1}\right)\left({k - 2}\right)}\right)} 2$

Thus: