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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PROOF WITHOUT WORDS: SUMS OF SUMS OF TRIANGULAR NUMBERS

Authors

  • Chuya Fukuda

Keywords:

triangular numbers, consecutive natural numbers, tetrahedron.

DOI:

https://doi.org/10.17654/0973563123002

Abstract

A pictorial proof of the Sums of Sums of Triangular Numbers, that is,

$$
4\left(T_1+\left(T_1+T_2\right)+\left(T_1+T_2+T_3\right) \cdots\left(T_1+\cdots+T_n\right)\right)=(n+3) \sum_{k=1}^n T_k,
$$

is provided. By expressing each triangular number as a sum of consecutive natural numbers, we can place those natural numbers into a tetrahedron.  And if we change the base of this tetrahedron four times, we get four tetrahedrons. By summing up the corresponding parts of these four tetrahedrons, the Sums of Sums of Triangular Numbers can be expressed as the formula shown above.

Received: December 2, 2022
Accepted: December 10, 2022

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Published

19-12-2022

Issue

Vol. 24 (2023)

Section

Articles

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Copyright (c) 2023 Far East Journal of Mathematical Education

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How to Cite

PROOF WITHOUT WORDS: SUMS OF SUMS OF TRIANGULAR NUMBERS. (2022). Far East Journal of Mathematical Education, 24, 5-6. https://doi.org/10.17654/0973563123002