Home | Business News | Browse by Publication | G | Georgia Journal of Science

The fascinating mathematical beauty of triangular numbers.

Article Excerpt
ABSTRACT

The triangular numbers are formed by partial sum of the series 1 + 2 + 3 + 4 + 5 + 6 + 7. ... + n (2). In other words, triangular numbers are those counting numbers that can be written as = 1 + 2 + 3 + ... + n. So.

[T.sub.1] = 1

[T.sub.2] = 1 + 2=3

[T.sub.3] = 1+2+3=6

[T.sub.4] = 1 + 2 + 3 + 4 = 10

[T.sub.5] = 1 + 2 + 3 + 4 + 5 = 15

[T.sub.6] = 1 + 2 + 3 + 4 + 5 + 6 = 21

[T.sub.7] = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

[T.sub.8] = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

[T.sub.9] = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

[T.sub.10] = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

In this paper we investigate some important properties of triangular numbers. Some important results dealing with the mathematical concept of triangular numbers will be proved. We try our best to give short and readable proofs. Most of the results are supplemented with examples.

Key Words: Triangular numbers, Perfect square, Pascal Triangles, and perfect numbers.

INTRODUCTION

The sequence 1, 3, 6, 10, 15, ..., n(n + 1)/2, ... shows up in many places of mathematics (1). The Greek called them triangular numbers (1). The triangular number [T.sub.n] is a figurate number that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one as shown below (2).

* * ** * ** *** * ** *** **** * ** *** **** *****...

View this article FREE - Now for a Limited Time, try Goliath Business News
Free for 3 Days!