Let f: N R be a function satisfying the relation _ k=1 ^x f(k) = x^2 f(x) - 2 for all integers x 2 . If _ x=1 ^ f(x) = 10 , then the value of f(1) is equal to

Step-by-step solution

Let f(1) = . Substitute x=2 into the given equation: f(1) + f(2) = 4f(2) - 2 + 2 = 3f(2) f(2) = + 2 3 For x 3 , let S_x = _ k=1 ^x f(k) . We have: S_x = x^2 f(x) - 2 S_ x-1 = (x-1)^2 f(x-1) - 2 Subtracting the second equation from the first: f(x) = x^2 f(x) - (x-1)^2 f(x-1) (x^2 - 1) f(x) = (x-1)^2 f(x-1) (x+1)f(x) = (x-1)f(x-1) Thus, f(x) = x-1 x+1 f(x-1) for x 3 . By telescoping products: f(x) = ( x-1 x+1 ) ( x-2 x ) ( 2 4 ) f(2) f(x) = 2 3 x(x+1) f(2) = 6 x(x+1) + 2 3 = 2( + 2) x(x+1) This formula is valid for x