VMO 2012 Problem 1

Define a sequence $\{x_n\}$ as: $\left\{\begin{aligned}& x_1=3 & x_n=\frac{n+2}{3n}(x_{n-1}+2)\ \ \text{for} \ n\geq 2.\end{aligned}\right.$ Prove that this sequence has a finite limit as $n\to+\infty.$ Also determine the limit.