Closed Form for Pentagonal Numbers

Theorem
The closed-form expression for the $n$th pentagonal number is:
 * $P_n = \dfrac {n \left({3 n - 1}\right)} 2$

Proof
Pentagonal numbers are $k$-gonal numbers where $k = 5$.

From Closed Form for Polygonal Numbers we have that:
 * $P \left({k, n}\right) = \dfrac n 2 \left({\left({k - 2}\right) n - k + 4}\right)$

Hence:

Hence the result.