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:
 * $\displaystyle P \left({k, n}\right) = \frac {n \left({2 + \left({n-1}\right)\left({k-2}\right)}\right)} 2$

Thus: