MathDB

Power of two sequence inequality

Source: IMO Shortlist 1995, S4

August 18, 2004

inequalitiesfunctionalgebrapolynomialIMO Shortlist

Problem Statement

Suppose that

x_1, x_2, x_3, \ldots

are positive real numbers for which x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n for n \equal{} 1, 2, 3, \ldots Prove that

\forall n,

2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.