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.

$\map P {k, n} = \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$

Thus:

$\ds \map P {k + 1, n} - \map P {k, n}$

$=$

$\ds \dfrac n 2 \paren {\paren {\paren {k + 1} - 2} n - \paren {k + 1} + 4} - \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$

$\ds $

$=$

$\ds \dfrac n 2 \paren {\paren {k - 1} n - k + 3} - \dfrac n 2 \paren {\paren {k - 2} n - k + 4}$

$\ds $

$=$

$\ds \dfrac n 2 \paren {k n - n - k + 3} - \dfrac n 2 \paren {k n - 2 n - k + 4}$

$\ds $

$=$

$\ds \dfrac n 2 \paren {k n - n - k + 3 - k n + 2 n + k - 4}$

$\ds $

$=$

$\ds \dfrac n 2 \paren {- n + 3 + 2 n - 4}$

$\ds $

$=$

$\ds \frac {n \paren {n - 1} } 2$

$\ds $

$=$

$\ds T_{n - 1}$

$\blacksquare$