1st Irish 1988 Problem 8 The sequence of nonzero reals x 1 , x 2 , x 3 , ... satisfies x n = x n-2 x n-1 /(2x n-2 - x n-1 ) for all n > 2. For which (x 1 , x 2 ) does the sequence contain infinitely many integral terms?
x 1 = x 2 = non-zero integer Solution Put a n = 1/x n . Then a n = 2a n-1 - a n-2 , so a n - a n-1 = a n-1 - a n-2 . Iterating, a n - a n-1 = a 2 - a 1 . Iterating, a n = a 1 + (n-1)(a 2 -a 1 ). Now if a 2 ≠ a 1 (or equivalently x 1 ≠ x 2 ), then for all but finitely many terms we have |a n | > 1 and hence |x n | < 1, and hence (since x n is non-zero) x n not integral. If x 1 = x 2 , then all x n = x 1 . Thanks to Suat Namli 1st Irish 1988 © John Scholes jscholes@kalva.demon.co.uk 12 February 2004 Last updated/corrected 12 Feb 04