# Closed Form for Pentagonal Numbers

## Theorem

The closed-form expression for the $n$th pentagonal number is:

$P_n = \dfrac {n \paren {3 n - 1} } 2$

## Proof

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

From Closed Form for Polygonal Numbers we have that:

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

Hence:

 $\ds P_n$ $=$ $\ds \frac n 2 \paren {\paren {5 - 2} n - 5 + 4}$ Closed Form for Polygonal Numbers $\ds$ $=$ $\ds \dfrac {n \paren {3 n - 1} } 2$

Hence the result.

$\blacksquare$