Pentagonal Number — Definition, Formula & Examples

A pentagonal number is a figurate number that counts the number of dots in a pattern forming nested pentagons. The sequence begins 1, 5, 12, 22, 35, 51, ...

The $n$ -th pentagonal number $p_n$ is defined by $p_n = \frac{n(3n-1)}{2}$ for positive integers $n$ . Equivalently, $p_n$ is the sum of the first $n$ terms of the arithmetic sequence $1, 4, 7, 10, \ldots$ whose common difference is 3.

Key Formula

p_n = \frac{n(3n - 1)}{2}

Where:

$p_n$ = The n-th pentagonal number
$n$ = A positive integer indicating position in the sequence

How It Works

To find the

n

-th pentagonal number, substitute

n

into the formula

\frac{n(3n-1)}{2}

. You can also build each pentagonal number from the previous one: add

3n - 2

to the

(n-1)

-th pentagonal number to get the

n

-th. Geometrically, each layer of the pentagon adds a new ring of dots around the previous figure, and that ring contains exactly

3n - 2

dots.

Worked Example

Problem: Find the 7th pentagonal number.

Substitute into the formula: Replace

n

with 7 in the pentagonal number formula.

p_7 = \frac{7(3 \cdot 7 - 1)}{2} = \frac{7 \cdot 20}{2}

Simplify: Multiply and divide.

p_7 = \frac{140}{2} = 70

Answer: The 7th pentagonal number is 70.

Visualization

Why It Matters

Pentagonal numbers appear in Euler's pentagonal number theorem, which provides a remarkable formula for integer partitions — a central topic in combinatorics and number theory. Generalized pentagonal numbers (allowing negative indices in the formula) serve as exponents in an infinite product expansion of

(1 - x)(1 - x^2)(1 - x^3)\cdots

, connecting figurate numbers to deep algebraic identities.

Common Mistakes

Mistake: Confusing triangular number and pentagonal number formulas.

Correction: Triangular numbers use

\frac{n(n+1)}{2}

, while pentagonal numbers use

\frac{n(3n-1)}{2}

. The coefficient 3 in the pentagonal formula reflects the pentagon's geometry.