Triangle numbers

Triangle numbers are numbers that start from one and increase in a pattern of triangles shown by the dots below.

The sequence starts with 1 dot and the number of dots increases by adding 1 dot to the base of the previous triangle.

Each triangle number is the previous number plus the number of yellow dots. The number of yellow dots increases by one with each new number. We can use this information to produce a recursive (which means occurring again) relationship between triangle numbers.

The number of dots in each triangle is given in the table below. If n = sequence number and is the number of dots for n then

n	1	2	3	4	5
	1	3	6	10	15

Notice that the number of yellow dots for n = 2 onwards in each triangle number is the same as n.

We can use this to produce our recursive relationship. and

Let’s test this

So,

We can also work out a formula to find out the number of dots for any n in the sequence without using a recursive relationship. To help us do this I have arranged the dots we have been using to represent triangle numbers in a different way.

If we double the number of dots in each picture, we get the following pattern

Each shape of dots above is a rectangle. Each rectangle has n dots at the bottom and n + 1 dots up the side

The number of dots in each rectangle is the base times the height which is .

The number of dots in each triangle is half of this, .

Let’s test this

We can also work out what the 50th term in the sequence is