Triangular Number Sequence

This is the Triangular Number Sequence:

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

It is simply the number of dots in each triangular pattern:

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

x₅ = 5(5+1)/2 = 15

Example: the 60th is

x₆₀ = 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?

x₂₂ = 22(22+1)/2 = 253

The stack may be dangerously high, but you can fit 253 logs in it!