How can you identify a triangular number?

A triangular number can be identified using the formula ( T_n = \frac{n(n + 1)}{2} ), where ( n ) is a positive integer. This formula represents the sum of the first ( n ) natural numbers. Additionally, a number ( x ) is triangular if the equation ( n^2 + n - 2x = 0 ) has an integer solution for ( n ), which can be determined using the quadratic formula. Alternatively, you can check if ( 8x + 1 ) is a perfect square, as this is another characteristic of triangular numbers.