k is a positive constant. The sequence x_{i} of positive reals has sum k. What are the possible values for the sum of x_{i}^{2} ?

**Solution**

Answer: any value in the open interval (0, k^{2}).

Since the terms are positive we have 0 < ∑ x_{n}^{2} < ( ∑ x_{n})^{2} = k^{2}. So certainly all sums must lie in the interval (0, k^{2}).

We show how to get any value in the interval by explicit construction. We may represent the value as hk^{2} 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), kc^{2}(1 - c), ... ). We have ∑ x_{n} = k as required, and ∑ x_{n}^{2} = k^{2}(1 - c)/(1 + c) = hk^{2} as required.

© John Scholes

jscholes@kalva.demon.co.uk

1 Jan 2001