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.

If a triangle number is expressed as 1+2+3+...+N, then its place is N. (1 is the place of 1, 2 is the place of 1+2, 3 is the place of 6=1+2+3, 4 is the place of 1+2+3+4. etc.)

1+2+3+...+N = N[(1+N)/2]

8(1+2+3+...+N)+1 = 8(N[(1+N)/2])+1
= 4N^2 + 4N + 1
= (2N+1)^2

Posted by Penny on 2004-01-15 09:34:21