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Prove that 45< x_1000 <45.1

Source:

September 21, 2010

integrationalgebraInequalityapproximationSequencerecurrence relationIMO Shortlist

Problem Statement

Let

x_0 = 5

and

x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )

. Prove that

45 < x_{1000} < 45. 1.