Let S = n Z : n^2 - 4n - 100 n^2 - 4n + 100 1 2 and f(x) = x^3 - 6x^2 + 13x - 5 . Then the value of _ n S f(n) is equal to _______.

Step-by-step solution

First, simplify the inequality defining the set S . Notice that the denominator is n^2 - 4n + 100 = (n-2)^2 + 96 , which is strictly greater than 0 for all real n . Since the denominator is positive, we can cross-multiply without changing the inequality sign: 2(n^2 - 4n - 100) n^2 - 4n + 100 2n^2 - 8n - 200 n^2 - 4n + 100 n^2 - 4n - 300 0 Completing the square for the quadratic: (n-2)^2 - 4 - 300 0 (n-2)^2 304 Since 17^2 = 289 and 18^2 = 324 , the integer values of (n-2) that satisfy this inequality are -17, -16, ,