 #### 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 nth square number, sn = n²

• Triangle numbers The nth triangle number, tn = ½n(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 nth triangle number, tn, is the sum of the first n natural numbers.
To find a formula for this we first consider, t100 = 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 t100 = 50 × 101 = 5050.

In the same way, tn = 1 + 2 + 3 + ... + n is equal to ½n pairs of (n + 1); confirming the result, tn = ½n(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) = 2t4, but if we arrange these two triangles to form a rectangle. We can see that 2t4 = 4 × 5 = 20, so t4 = ½ × 20 = 10.

Generally, 2tn = n(n + 1), hence tn = ½n(n + 1).