Sum of the powers of a natural number is proved pictorially via a one-dimensional modification of the two- and three-dimensional representations proposed by Mukherjee in a past study. As examples, sum of the powers of 4 and sum of the powers of 8 are illustrated by drawing an array of rectangles. These diagrams can be regarded as reflecting the self-similarity of the geometric structures of the above-mentioned sums.

sum of the powers of a natural number, proof without words, self-similarity.

Received: April 29, 2023; Accepted: May 19, 2023; Published: June 5, 2023

How to cite this article: Yukio Kobayashi, Geometric structure of sum of the powers of a natural number, Far East Journal of Mathematical Education 24 (2023), 37-44. http://dx.doi.org/10.17654/0973563123010

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

References:

[1] Rajib Mukherjee, Proof without words: Sums of powers of eight, The Mathematical Intelligencer 43 (2001), 60-61.
[2] David B. Sher, Sums of powers of four, Mathematics and Computer Education 31 (1997), 190.
[3] Yukio Kobayashi, Proof without words: A sum computed by self-similarity, College Mathematics Journal 49 (2018), 10.
[4] Mingjang Chen, Proof without words: Sums of consecutive powers of n via self-similarity, Math. Mag. 77 (2004), 373.
[5] Yukio Kobayashi, A pedagogical study of geometric relations between the sums of cubic integer numbers and the sums of square integer numbers with the sums of odd numbers, European Journal of Pure and Applied Mathematics 15 (2022), 864-877.
[6] Yukio Kobayashi, Relations among powers of 2, combinations, and symbolic algebra, Mathematics Teacher 99 (2006), 577-578.
[7] Sanjay Kumar Khattri, Proof without words: Teaching Mathematics and its Applications: An International Journal of the IMA 27 (2008), 220-222.