Extension: Triangular Numbers

The triangular numbers are a useful sequence of numbers, named after the fact that they can be arranged in a triangle as follows:

One way to find the $n^{th}$ triangular number is by using the following formula: $$T_n = \frac{n(n+1)}{2}$$ Triangular numbers have applications beyond arranging circles into triangles, of course; for example, consider the following identity: $$1 + 2 + \dots + n = \sum_{i=1}^n i = T_n$$ This can be easily seen using the picture above, as each row of the triangle has $i$ circles in it, for $i = 1$ to $n$.

Here are a few more interesting identities involving the triangular numbers: $$\sum_{i=1}^n i^3 = T_n^2$$ $${n+1 \choose 2} = T_n$$ $$T_{n+1}^2 - T_n^2 = (n+1)^3$$

Triangular Numbers

We have learned about natural numbers and and perfect numbers. We've also learned about square numbers, like $4$ and $16$. You may also be familiar with hexagonal numbers, like $1$, $6$, $15$, and $28$ (number of dots around a hexagon). The next lesson pertaining to figurate numbers is triangular numbers!

We previously learned about the finite arithmetic series. An extension of this principle is to calculate Triangular Numbers.

There are many uses for triangular numbers, specifically finding the binomial coefficient.

(Video) Triangular Numbers Explained

by Minity Maths

In this video by Minity Maths, you will get an explanation for what triangular numbers are and the patterns associated with them.