IMO 1983 LL KUW38

Origin: KUW

Problem

Let {un} be the sequence defined by its first two terms u0, u1 and the recursion formula un+2 = un −un+1. (a) Show that un can be written in the form un = \alphaan + \betabn, where a, b, \alpha, \beta are constants independent of n that have to be determined. (b) If Sn = u0 + u1 + \cdot \cdot \cdot + un, prove that Sn + un−1 is a constant inde- pendent of n. Determine this constant.