For every real number x_{1}, construct the sequence x_{1}, x_{2}, ... by setting:

x_{n+1} = x_{n}(x_{n} + 1/n).

Prove that there exists exactly one value of x_{1} which gives 0 < x_{n} < x_{n+1} < 1 for all n.

**Solution**

Define S_{0}(x) = x, S_{n}(x) = S_{n-1}(x) (S_{n-1}(x) + 1/n). The motivation for this is that x_{n} = S_{n-1}(x_{1}).

S_{n}(0) = 0 and S_{n}(1) > 1 for all n > 1. Also S_{n}(x) has non-negative coefficients, so it is strictly increasing in the range [0,1]. Hence we can find (unique) solutions a_{n}, b_{n} to S_{n}(a_{n}) = 1 - 1/n, S_{n}(b_{n}) = 1.

S_{n+1}(a_{n}) = S_{n}(a_{n}) (S_{n}(a_{n}) + 1/n) = 1 - 1/n > 1 - 1/(n+1), so a_{n} < a_{n+1}. Similarly, S_{n+1}(b_{n}) = S_{n}(b_{n}) (S_{n}(b_{n}) + 1/n) = 1 + 1/n > 1, so b_{n} > b_{n+1}. Thus a_{n} is an increasing sequence and b_{n} is a decreasing sequence with all a_{n} less than all b_{n}. So we can certainly find at least one point x_{1} which is greater than all the a_{n} and less than all the b_{n}. Hence 1 - 1/n < S_{n}(x_{1}) < 1 for all n. But S_{n}(x_{1}) = x_{n+1}. So x_{n+1} < 1 for all n. Also x_{n} > 1 - 1/n implies that x_{n+1} = x_{n}(x_{n} + 1/n) > x_{n}. Finally, we obviously have x_{n} > 0. So the resulting series x_{n} satisfies all the required conditions.

It remains to consider uniqueness. Suppose that there is an x_{1} satisfying the conditions given. Then we must have S_{n}(x_{1}) lying in the range 1 - 1/n, 1 for all n. [The lower limit follows from x_{n+1} = x_{n}(x_{n} + 1/n).] Hence we must have a_{n} < x_{1} < b_{n} for all n. We show uniqueness by showing that b_{n} - a_{n} tends to zero as n tends to infinity. Since all the coefficients of S_{n}(x) are non-negative, it is has increasing derivative. S_{n}(0) = 0 and S_{n}(b_{n}) = 1, so for any x in the range 0, b_{n} we have S_{n}(x) ≤ x/b_{n}. In particular, 1 - 1/n < a_{n}/b_{n}. Hence b_{n} - a_{n} ≤ b_{n} - b_{n}(1 - 1/n) = b_{n}/n < 1/n, which tends to zero.

Solutions are also available in Murray S Klamkin, International Mathematical Olympiads 1978-1985, MAA 1986, and in István Reiman, International Mathematical Olympiad 1959-1999, ISBN 189-8855-48-X.

© John Scholes

jscholes@kalva.demon.co.uk

22 Oct 1998