The visual clue for the series: sum of the product of two consecutive integers is presented. The diagrammatic proof of this sum can also be used to prove the formula for the sum of square integers. It can be understood from the diagram that (n + 1)n is the common factor, and thus the relationship between the sum of natural numbers, the sum of square integers, and the sum of the product of two consecutive integers explains why each of the above series,   and  is represented by the same factor of

sum of the product of two consecutive integers, sum of natural numbers, sum of square number integers, proof without words.

Received: November 18, 2022; Accepted: December 10, 2022; Published: February 17, 2023

How to cite this article: Yukio Kobayashi, Proof without words: The sum of the product of two consecutive integers, Far East Journal of Mathematical Education 24 (2023), 17-20. http://dx.doi.org/10.17654/0973563123007

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

References:

[1] Yukio Kobayashi, Proof without words: Relationship between the sum of natural numbers and the sum of the product of two consecutive integers, Far East J. Math. Edu. 22 (2022), 1-3.
[2] Roger G. Nelsen, Proof without words II: More exercises in visual thinking, The Mathematical Association of America, 2000, p.105.
[3] Yukio Kobayashi, Proof without words: Relationship between the sum of triangular numbers, sum of natural numbers, and sum of square numbers, Far East J. Math. Edu. 21 (2021), 83-91.