# Triangular Number Sequence

This is the Triangular Number Sequence:

1, 3, 6, 10, 15, 21, 28, 36, 45, ... |

This sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

## A Rule

We can make a "Rule" so we can calculate any triangular number.

First, rearrange the dots (and give each pattern a number *n*), like this:

Then double the number of dots, and form them into a rectangle:

- The rectangles are
*n*high and*n+1*wide - and
**x**is how many dots in the triangle (the value of the Triangular Number_{n}*n*)

And we get (remembering we doubled the dots):

2x

_{n}= n(n+1)x

_{n}= n(n+1)/2Rule: x_{n} = n(n+1)/2

Example: the **5th** Triangular Number is

x_{5} = 5(5+1)/2 = **15**

Example: the **60th** is

x_{60} = 60(60+1)/2 = **1830**

Wasn't it much easier to use the formula than to add up all those dots?