Mock AIME 3 Pre 2005 Problems/Problem 15
Problem
Let denote the value of the sum
The value of can be expressed as , where and are relatively prime positive integers. Compute .
Solution
Let
Factoring the radicand, we have The fraction looks remarkably apt for a trigonometric substitution; namely, define such that . Then the RHS becomes But Therefore, This gives us So now When we sum , this sum now telescopes: Therefore, the required value giving us the desired answer of .