PROOF WITHOUT WORDS: THE SUM OF THE PRODUCT OF TWO CONSECUTIVE INTEGERS

Abstract

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, $1+2+\cdots+n$, $1^2+2^2+\cdots+n^2$, and $2 \cdot 1+3 \cdot 2+\cdots+(n+1) n$, is represented by the same factor of $(n+1) n$.

Received: November 18, 2022 
Accepted: December 10, 2022