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Sequence from Iceland

Source: IMO LongList 1988, Iceland 2, Problem 35 of ILL

October 22, 2005
inductionlimitalgebra unsolvedalgebra

Problem Statement

A sequence of numbers an,n=1,2,…,a_n, n = 1,2, \ldots, is defined as follows: a1=12a_1 = \frac{1}{2} and for each n≥2n \geq 2 an=2n−32nan−1. a_n = \frac{2 n - 3}{2 n} a_{n-1}. Prove that ∑k=1nak<1\sum^n_{k=1} a_k < 1 for all n≥1.n \geq 1.
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