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.

Average Figurate Numbers

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  Written for educators - and their students too! - this website, slowly growing, takes all the content Tanton has developed in his books, videos, and workshops, and organizes it into short, self-contained, and complete, curriculum units proving that mathematics, at all points of the school curriculum, can be joyous, fresh, innovative, rich, deep-thinking, and devoid of any rote doing! Let's teach generations of students to be self-reliant thinkers, willing to flail and to use their common sense to "nut their way" through challenges, to assess and judge results, and to adjust actions to find success. (Great life skills!)

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• Books

  • Thinking Mathematics! (TEN VOLUMES)
  • Solve This: Mathematical Activities for Students and Clubs
  • Encyclopedia of Mathematics
  • Geometry I & II

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