7th CanMO 1975 Problem 2 Define the real sequence a 1 , a 2 , a 3 , ... by a 1 = 1/2, n 2 a n = a 1 + a 2 + ... + a n . Evaluate a n .
We show by induction that a n = 1/(n(n+1) ). Obviously true for n = 1. Suppose it is true for n. We need to show that (n+1) 2 /( (n+1)(n+2) ) - n 2 /( n(n+1) ) = 1/( (n+1)(n+2) ), or (n+1)/(n+2) - n/(n+1) = 1/( (n+1) (n+2) ), or (n+1) 2 - n(n+2) = 1, which is true. 7th CanMO 1975 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002