A function f: R R satisfies the relation f(x+y) = f(x) + f(y) + 3xy - 2 for all x, y R . If f(1) = 5 , then the value of _ n=1 ¹⁰ f(n) is equal to:

Step-by-step solution

Given the functional equation: f(x+y) = f(x) + f(y) + 3xy - 2 Substitute x = 0 and y = 0 : f(0) = f(0) + f(0) + 0 - 2 f(0) = 2 Now, substitute y = 1 in the given equation: f(x+1) = f(x) + f(1) + 3x(1) - 2 Since f(1) = 5 , we have: f(x+1) = f(x) + 5 + 3x - 2 f(x+1) - f(x) = 3x + 3 Replacing x with n-1 where n is an integer: f(n) - f(n-1) = 3(n-1) + 3 = 3n Summing this relation from k = 1 to n : _ k=1 ^ n (f(k) - f(k-1)) = _ k=1 ^ n 3k f(n) - f(0) = 3 n(n+1) 2 f(n) = 3 2 n(n+1) + 2 We need to find _ n=1 ¹⁰ f(n) : _ n