## Frequently Asked Questions

#### What are square and triangle numbers?

If you are given `n` tiles and you can form a perfect figure with them, then `n` belongs to the family of figurate or polygonal numbers. Square and triangle numbers are the most common sets of figurate numbers, but there are pentagonal, hexagonal, heptagonal, et cetera.

__Square numbers__

The

`n`th square number,`s`=_{n}`n`²__Triangle numbers__

The

`n`th triangle number,`t`= ½_{n}`n`(`n`+ 1).

They appear in many familiar contexts. Think about the shape and the number of pins in the game 10-pin bowling, or the way in which the fifteen red balls in a game of snooker are placed.

You may notice that the `n`th triangle number, `t _{n}`, is the sum of the first

`n`natural numbers.

To find a formula for this we first consider,

`t`

_{100}= 1 + 2 + 3 + ... + 100.

As 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, ... , 50 + 51 = 101, we observe that the sum, 1 + 2 + 3 + ... + 100, is equal to fifty pairs of 101. Hence `t`_{100} = 50 × 101 = 5050.

In the same way, `t _{n}` = 1 + 2 + 3 + ... +

`n`is equal to ½

`n`pairs of (

`n`+ 1); confirming the result,

`t`= ½

_{n}`n`(

`n`+ 1).

We can show this result by a different method. Consider the diagram.

Interestingly this shows that 2 + 4 + 6 + 8 = 2(1 + 2 + 3 + 4) = 2`t`_{4}, but if we arrange these two triangles to form a rectangle.

We can see that 2`t`_{4} = 4 × 5 = 20, so `t`_{4} = ½ × 20 = 10.

Generally, 2`t _{n}` =

`n`(

`n`+ 1), hence

`t`= ½

_{n}`n`(

`n`+ 1).