Using the Sum of Triangular Numbers as the most Fundamental Number of Combinations Found using Heuristic Methods

Abstract

The same mathematical meaning can be expressed in many different ways. Students should deduce this and thus perceive that the different expressions have the same meaning. In this work, a pedagogical approach to the determination of various equations related to the sum of triangular numbers is presented as an example to show how students can deduce these different expressions and their meanings. The sum of triangular numbers is closely related to the sums of natural numbers, square numbers, and two consecutive numbers. These relationships can be found using a heuristic method in which the equations representing the sum of natural numbers are simply listed. Guiding students to discover the formula by themselves is important, rather than giving them the formula from the beginning and asking them to prove it. Triangular numbers are the fundamental numbers of combinations. The sum of $n$ consecutive triangular numbers is equal to ${}_{n+2}{C}_3$, as shown by combinatoric tree diagrams. The sum of the triangular numbers can be regarded as an extended version of the sum of the natural numbers.  The sum of square numbers is also related to the sum of triangular numbers, which is also easy to understand in terms of combinatorics. The combinatoric expression for the sum of triangular numbers can be extended to ${\displaystyle \sum _{\ell =1}^{k-1}{}_{n-\ell }{C}_{k-1}={}_n{C}_k}$, which can be expressed as the sum of triangular numbers. This study provides an example of teaching material that broadens students’ view of a given equation by crossing different learning units such as combinatorics and algebra. The sum of triangular numbers discussed in this study is simply an example of developing the habit of considering various interpretations of the same formula, but even well known formulas can be utilized for this type of subject matter.