VN-TST 1985 Problem 1

The sequence $(x_n)$ of real numbers is defined by $x_1=\frac{29}{10}$ and $x_{n+1}=\frac{x_n}{\sqrt{x_n^2-1}}+\sqrt{3}$ for all $n\ge 1$ . Find a real number $a$ (if exists) such that $x_{2k-1}>a>x_{2k}$ .