ILL 1983 Problem 39

If $\alpha$ is the real root of the equation $$E(x) = x^3 - 5x -50 = 0$$ such that $x_{n+1} = (5x_n + 50)^{1/3}$ and $x_1 = 5$ , where $n$ is a positive integer, prove that:(a) $x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)$ (b) $\alpha < x_{n+1} < x_n.$