Two formulae for the sum of triangular numbers can be found by observing simple diagrams with squares arranged. These formulae indicate that the sum of triangular numbers is related to both the sum of natural numbers and sum of square numbers. The sum of triangular numbers can also be obtained from observing other diagrams with squares arranged. The sum of square numbers can be calculated by using both the sum of natural numbers and the sum of triangular numbers. Using pictures, we can find the relationship between these sums.

triangular numbers, sum of natural numbers, sum of square numbers, proof without words, visual thinking.

Received: March 12, 2021; Accepted: April 9, 2021; Published: May 26, 2021

How to cite this article: Yukio Kobayashi, Proof without words: relationship between the sum of triangular numbers, sum of natural numbers, and sum of square numbers, Far East Journal of Mathematical Education 21(1) (2021), 83-91. DOI: 10.17654/ME021010083

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

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