VN-TST 1990 Problem 2

Let be given four positive real numbers $a$ , $b$ , $A$ , $B$ . Consider a sequence of real numbers $x_1$ , $x_2$ , $x_3$ , $\ldots$ is given by $x_1 = a$ , $x_2 = b$ and $x_{n + 1} = A\sqrt [3]{x_n^2} + B\sqrt [3]{x_{n - 1}^2}$ ( $n = 2, 3, 4, \ldots$ ). Prove that there exist limit $\lim_{n\to + \propto}x_n$ and find this limit.