Is 190 a triangular number

Answer 1

The explicit formula of a triangular number is:

(n + 1 choose 2) which is equivalent to (n + 1)!/(2!(n + 1 - 2)!) = (n + 1)!/(2!(n - 1)!)

Assume that 190 is a triangular number. Then:

190 = (n + 1)!/(2!(n - 1)!)

190 = n(n + 1)/2

380 = n(n + 1)

380 = n² + n

0 = n² + n - 380

0 = (n + 20)(n - 19)

So n = -20 and n = 19

But n has to be positive since n determines the nth triangular number. Therefore, n = 19.

Therefore, 19 is a triangular number.