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.

The sequence of triangle numbers is (i)(i+1)/2
Multiplying this by 8 then adding 1 equals:
4i(i+1) +1 = 4i^2 + 4i +1 = (2i+1)^2

Posted by Larry on 2004-01-28 20:18:12