IMO 1983 LL KUW39

Origin: KUW

Problem

If \alpha is the real root of the equation E(x) = x3 −5x −50 = 0 such that xn+1 = (5xn + 50)1/3 and x1 = 5, where n is a positive integer, prove that: (a) x3 n+1 −\alpha3 = 5(xn −\alpha) (b) \alpha < xn+1 < xn