Triangle numbers are calculated taking each integer plus all the ones before it. The first triangle number is 1, the second is 1+2 or 3, and the third is 1+2+3 or 6.

If you take 8 times a triangle number plus 1, the result will be a perfect square. This number also will be the square of the triangle number's place doubled, plus one.

For example, 6 is third in the triangle number sequence. (1, 3, 6...) This means 8 times 6 plus 1 = 49 equals 3 times 2 plus 1, squared, or 7 squared.

Prove why this works.

each triangle number "k" is the sum of all natural nos from 1 to n.
this is the sum of an arithmatic progression from 1 to n with commen difference 1
k= n/2*(1*2+(n-1)1)=n(n+1)/2
8*the square of n +1 = 4(n^2+n)+1
which is a perfect square =(2n+1)^2
which is the square of the numbers place n doubled plus 1.

Posted by bharath on 2004-01-15 13:29:10