User:Wguynes/Proofs/Triangular number test

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Let T = number to test

T = \frac{n(n+1)}{2}

n2 + n − 2T = 0

Find the roots of the quadratic. What value of n would make our T?

n = \frac { -1 \pm \sqrt{ 1^2 - 4 (1)(-2T) } } { 2 \cdot 1 } = \frac{ -1 \pm \sqrt{ 8T + 1 } }{2}

Only one of these two roots can be a positive result.

n = \frac{ \sqrt{ 8T + 1 } - 1 }{2}