Difference Between Adjacent Polygonal Numbers is Triangular Number

Theorem
Let $\map P {k, n}$ be the $n$th $k$-gonal number.

Then:
 * $\map P {k + 1, n} - \map P {k, n} = T_{n - 1}$

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

Proof
From Closed Form for Polygonal Numbers:
 * $\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$

Thus: