Let f: R R be a polynomial function such that _ i=1 ^n f(i) = n(n+1)(2n+1) 6 - 2n for all n N . If f(x) satisfies the relation f(x+y) = f(x) + f(y) + x y + for all x, y R , then the value of 10 - is

Step-by-step solution

Let S_n = _ i=1 ^n f(i) = n(n+1)(2n+1) 6 - 2n . We know the standard summation formulas: _ i=1 ^n i^2 = n(n+1)(2n+1) 6 and _ i=1 ^n 2 = 2n . Therefore, the sum can be rewritten as: S_n = _ i=1 ^n (i^2 - 2) This implies that f(n) = n^2 - 2 for all natural numbers n . Since f(x) is a polynomial function, this identity must hold for all real numbers, so f(x) = x^2 - 2 . Substitute f(x) into the given functional equation: f(x+y) = f(x) + f(y) + x y + (x+y)^2 - 2 = (x^2 - 2) + (y^2 - 2) + x y + x^2 + y^2 + 2xy - 2 = x^2