PROOF WITHOUT WORDS: RELATIONSHIP AMONG THE SUM OF CUBIC INTEGER NUMBERS, SUM OF CONSECUTIVE ODD NUMBERS, AND SUM OF NATURAL NUMBERS

Abstract

A pictorial proof of the relationship among the sum of cubic integer numbers, sum of consecutive odd numbers, and sum of natural numbers, that is $\sum_{k=1}^n k^3=\sum_{\ell=1}^{(1+n) n / 2}(2 \ell-1)=\left(\sum_{k=1}^n k\right)^2$, is provided. By expressing the sum of cubic integer numbers in terms of the number of squares and repeating the rearrangement of squares based on the idea of isoperimetric transformation, it can be understood that the sum of cubic integer numbers can be expressed as the sum of consecutive odd numbers as well as the square of the sum of natural numbers.

Received: October 19, 2022 
Accepted: November 15, 2022