Everyone
knows what squares are. The product of a number *n* with itself
is a square, so called because you can make an *n *×
*n* array from *n ^{2}*
objects. Few people, however, know about triangle numbers. Triangle
numbers result from arranging objects to form a triangle.

Thus,

It should also be
apparent that each *T _{n}
*is the sum of the first

Now suppose *n*
is even. Then we can rearrange the terms of the sum by pairing
the first term with the last, second term with the second last, and
so on to get,

The result is
terms
all equal to (*n *+ 1).
Thus,

This result can also be
proved by mathematical induction.
Similarly, if *n*
is odd, we get
terms all equal to (*n* +
1) plus an additional term
.
This also reduces to,

Triangle numbers have a number of interesting properties. For example, the sum of two adjacent triangle numbers is always a perfect square. You can easily see this from the illustration.

You
can also prove it by considering: