Concept 13 of 19Advanced

Pentagonal, hexagonal & other figurate numbers

Beyond triangles and squares, dots can form any regular polygon:

Pentagonal numbers: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145 … Formula: Pₙ = n(3n−1)/2
Hexagonal numbers: 1, 6, 15, 28, 45, 66, 91, 120 … Formula: Hₙ = n(2n−1)
Heptagonal (7-gon): 1, 7, 18, 34, 55, 81, 112 …

Stunning connection: every hexagonal number is also a triangular number: Hₙ = T_{2n−1}.

And: sums of these polygonal numbers build 3D "pyramid" versions (like tetrahedral for triangular).

Example

20th pentagonal = 20·(3·20−1)/2 = 20·59/2 = 590.
Is 15 hexagonal? H₃ = 3·5 = 15 ✓. Also H₃ = T₅ = 15 ✓ (same number!).

💡 Tip:All these formulas have the form n(an−b)/2. Once you see the pattern, they're easy to recall.

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