Far East Journal of Mathematical Education

The Far East Journal of Mathematical Education is a peer-reviewed journal focused on mathematical education. It publishes research papers that enhance understanding of mathematical concepts and encourages the use of technology, statistics, algorithms, and simulations in mathematics learning.

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SOLID GEOMETRICAL RELATIONSHIP AMONG THE SUMS OF INTEGERS, SQUARES, AND CUBES

Authors

  • Yukio Kobayashi

Keywords:

sum of integers, sum of squares, sum of cubes, volume of square prism

DOI:

https://doi.org/10.17654/0973563125006

Abstract

The relationship among the sum of integers, sum of squares, and sum of cubes is illustrated visually from a solid geometrical standpoint. These sums satisfy a characteristic relational equation,

$$
\sum_{k=1}^n k \times \sum_{k=1}^n k^2 / \sum_{k=1}^n k^3=(2 n+1) / 3 .
$$


This formula expresses the relationship between the base area and height of a square prism. The relationships among these sums are consistent with the dimensional analysis.

Received: March 16, 2025
Accepted: March 29, 2025

References

Yukio Kobayashi, Geometrical meaning of arithmetic series and in terms of the elementary combinatorics, International Journal of Mathematical Education in Science and Technology 42 (2011), 657-664. [Corrigendum Vol. 42 (2011), 1123.]

Yukio Kobayashi, Recursion formulae of and their combinatoric meaning in terms of the tree diagrams, International Journal of Mathematical Education in Science and Technology 44 (2013), 132-142.

Félix Martínez de la Rosa, Proof without words: sums of cubes in puzzle form, Far East Journal of Mathematical Education 23 (2022), 7.

Roger G. Nelsen, Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials), The Mathematical Association of America, 1997.

Yukio Kobayashi, Proof without words: Relationship among the sum of cubic integer numbers, sum of consecutive odd numbers, and sum of natural numbers, Far East Journal of Mathematical Education 24 (2023), 1-3.

Yukio Kobayashi, Supplementary remarks on a weak proof of the Pythagorean theorem - from the viewpoint of dimensional analysis of physical quantities, Far East Journal of Mathematical Education 27(1) (2025), 1-10.

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Published

06-05-2025

Issue

Vol. 27 No. 1 (2025)

Section

Articles

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How to Cite

SOLID GEOMETRICAL RELATIONSHIP AMONG THE SUMS OF INTEGERS, SQUARES, AND CUBES. (2025). Far East Journal of Mathematical Education, 27(1), 51-60. https://doi.org/10.17654/0973563125006

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