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Real number between two subsequences

Source: Vietnam TST 1985 Problem 1

February 5, 2010

algebra proposedalgebra

Problem Statement

The sequence

(x_n)

of real numbers is defined by x_1\equal{}\frac{29}{10} and x_{n\plus{}1}\equal{}\frac{x_n}{\sqrt{x_n^2\minus{}1}}\plus{}\sqrt{3} for all

n\ge 1

. Find a real number

a

(if exists) such that x_{2k\minus{}1}>a>x_{2k}.