45th Putnam 1984 Problem A2 Evaluate 6/( (9 - 4)(3 - 2) ) + 36/( (27 - 8)(9 - 4) ) + ... + 6 n /( (3 n+1 - 2 n+1 )(3 n - 2 n ) ) + ... . Solution
: 2. An easy induction on n shows that the sum of the first n terms is 6(3 n - 2 n )/(3 n+1 - 2 n+1 ). We may write this as 2 (1 - (2/3) n )/(1 - (2/3) n+1 ), which shows that it tends to 2 as n tends to infinity. 45th Putnam 1984 © John Scholes jscholes@kalva.demon.co.uk 7 Jan 2001