# Triangular Number Sequence

This is the Triangular Number Sequence:

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

This sequence comes from a pattern of **dots that form a
triangle**:

By adding another row of dots and counting all the dots we can

find
the next number of the sequence.

- The first triangle has just one dot.
- The second triangle has another row with 2 extra dots, making 1
**+ 2**= 3 - The third triangle has another row with 3 extra dots, making 1 + 2
**+ 3**= 6 - The fourth has 1 + 2 + 3
**+ 4**= 10 - etc!

*How may dots in the 60th triangle?*

## A Rule

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

First, rearrange the dots like this:

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

*Now* it is easy to work out how many dots: just multiply **n** by **n+1**

Dots in rectangle = n(n+1)

But remember we doubled the number of dots, so

Dots in triangle = n(n+1)/2

We can use x_{n} to mean "dots in triangle n", so we get the rule:

Rule: 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?

### Example: You are stacking logs.

There is enough ground for you to lay 22 logs side-by-side.

How many logs can you fit in the stack?

x_{22} = 22(22+1)/2 = **253**

You may get tired, and the stack may be dangerously high, but you can fit 253 logs in it!