Difference Between Adjacent Polygonal Numbers is Triangular Number

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

Then $$P \left({k+1, n}\right) - P \left({k, n}\right) = T_n$$, where $$T_n$$ is the $$n$$th triangular number.

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

Thus:

$$ $$ $$ $$ $$ $$