Relationships between pentagonal and triangular numbers

Relationships between pentagonal and triangular numbers figure 22016 is a triangular number.
The first three triangular numbers are: 1, 3, 6. The first three pentagonal numbers are: 1, 5, 12.

The pentagonal numbers 1, 5 and 12 are all one third of a triangular number.
Are all pentagonal numbers one third of a triangular number?

Reveal Solution

Algebra solution

The formula of the nth pentagonal number:
p_n=\frac{3n^2-n}{2}.

    \begin{align*} 3\left(\frac{3n^2-n}{2}\right)&=\frac{9n^2-3n}{2}\\ &=\frac{9n^2-6n+1-3n+6n-1}{2}\\ &=\frac{(3n-1)^2+(3n-1)}{2}. \end{align*}

i.e. the (3n-1)th triangular number.

Geometric solution

Relationships between pentagonal and triangular numbers figure 1

Problem Page Coordinator: Stephen Lee CMath MIMA – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today December 2016
Image credit: Pentagonal Plunge by Ken / Flickr / CC-BY-SA-2.0

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