k is a positive constant. The sequence xi of positive reals has sum k. What are the possible values for the sum of xi2 ?
Answer: any value in the open interval (0, k2).
Since the terms are positive we have 0 < ∑ xn2 < ( ∑ xn)2 = k2. So certainly all sums must lie in the interval (0, k2).
We show how to get any value in the interval by explicit construction. We may represent the value as hk2 where 0 < h < 1. Take c = (1 - h)/(1 + h). Take the geometric sequence with first term k(1 - c) and ratio c ( k(1 - c), kc(1 - c), kc2(1 - c), ... ). We have ∑ xn = k as required, and ∑ xn2 = k2(1 - c)/(1 + c) = hk2 as required.
61st Putnam 2000
© John Scholes
1 Jan 2001