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.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $22$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): pentagonal number
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): pentagonal number