PROOF WITHOUT WORDS: SUMS OF SUMS OF TRIANGULAR NUMBERS

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