SMO 2009 Open Round 1 Question 24

Let $(x_n)_{n=1}^{\infty}$ be real numbers with $x_1=3$, $x_2=24$, and $x_{n+2}=\tfrac14x_{n+1}+\tfrac34x_n$ for every positive integer $n$. Determine $\displaystyle\lim_{n\to\infty}x_n$.