1,3,6,10,15,21,28.

Somebody tells you a random positive integer of any size. Is there a quick way to work out if it is a triangle number or not (in other words, without going through all of the triangle numbers untill you get to a number as high or higher as the integer you've been told)?

Let the random positive integer be P.

If P is a triangle number, then by definition P = m(m+1)/2, for some positive integer m.

Now, we observe that:

8*P + 1
= 8{m(m+1)/2} + 1
= 4*m^2 + 4*m + 1
= (2*m + 1)^2

Thus, if (8*P + 1) is a perfect square, we will know for certain that P is a triangular number.