# Average Triangular, Square and Pentagonal Numbers (ESSAY!)

Which square numbers are the average of two other square numbers? 25, for example, is the average of 1 and 49, and 169 is the average of 49 and 289. Any more? Infinitely many more?

And what about triangular numbers that are the average of other triangular numbers? Or pentagonal numbers that are averages of other pentagonals?

This essay shows how to use ordinary Pythagorean triples: a^2 + b^2 = c^2 (of which there are infinitely many of those) to address these questions.