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:

triangular numbers

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

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:

triangular numbers 1 to 5

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

triangular numbers when doubled become n by n+1 rectangles

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 xn to mean "dots in triangle n", so we get the rule:

Rule: xn = 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?

log stack

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!