The sequence u0, u1, u2, ... is defined by: u0= 2, u1 = 5/2, un+1 = un(un-12 - 2) - u1 for n = 1, 2, ... . Prove that [un] = 2(2n - (-1)n)/3, where [x] denotes the greatest integer less than or equal to x.
Solution
Experience with recurrence relations suggests that the solution is probably the value given for [un] plus its inverse. It is straightforward to verify this guess by induction.
Squaring un-1 gives the sum of positive power of 2, its inverse and 2. So un-1 - 2 = the sum of a positive power of 2 and its inverse. Multiplying this by un gives a positive power of 2 + its inverse + 2 + 1/2, and we can check that the power of 2 is correct for un+1.
Solutions are also available in: Samuel L Greitzer, International Mathematical Olympiads 1959-1977, MAA 1978, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.
© John Scholes
jscholes@kalva.demon.co.uk
10 Oct 1998