A pictorial proof of the Sums of Sums of Triangular Numbers, that is,

is provided. By expressing each triangular number as a sum of consecutive natural numbers, we can place those natural numbers into a tetrahedron.  And if we change the base of this tetrahedron four times, we get four tetrahedrons. By summing up the corresponding parts of these four tetrahedrons, the Sums of Sums of Triangular Numbers can be expressed as the formula shown above.

triangular numbers, consecutive natural numbers, tetrahedron.

Received: December 2, 2022;  Accepted: December 10, 2022; Published: December 19, 2022

How to cite this article: Chuya Fukuda, Proof without words: sums of sums of triangular numbers, Far East Journal of Mathematical Education 24 (2023), 5-6. http://dx.doi.org/10.17654/0973563123002

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License