The sum of the first n positive integers is equal to 1/2(n squared+n). How many of the first n positive integers must be added to give a sum of 325?

The required equation is, 325 = ¹/₂(n² + n)...then 650 = n² + n...and the equation can be written:-
n² + n - 650 = 0...which can be factored
(n + 26)(n - 25) = 0
As we are only concerned with a positive integer solution, this occurs when n - 25 = 0, thus n = 25
Check : ¹/₂(25² + 25) = ¹/₂(625 + 25) = 325