I noticed your response only just now. I take it you’re looking for a general method to calculate the inverse of any function involving some kind of summation.
Short answer
There is no such general method. Just like there is no general method to solve any equation, or set of equations.
Slightly longer answer
For some summations the inverse can actually be analytically calculated. In your example, y = 1 + 2 + … + x = (x + 1) . x / 2, and reversing this is equivalent to solving x in the following quadratic equation: x2 + x - 2y = 0.
If an analytical approach isn’t available, you may try a numerical approach, or trial-and-error.