Argue why the function f: R rightarrow R defined by f(x) = x^3 - x^2 + x is invertible. Guess a closed form for the sum F_1 + F_3 + ... + F_2n - 1, where n > 0 is a natural number and (F_k)_k greaterthanorequalto 0 are Fibonacci numbers. This closed form may involve Fibonacci numbers. Show by induction that your guess is correct. Find a closed form for the double sum: sigma_1 lessthanorequalto j lessthanorequalto k lessthanorequalto n Attempt to find a closed form for the sum sigma_k = 1^n k^3 by perturbation, only to find a closed form for the following sum sigma_k = 1^n k^2