Convex combinations sequence question

Let $a_0,a_1,\beta$ be given with $0<\beta<1$ Let the sequence be defined by

$a_{n+2} = \beta a_{n+1} + (1-\beta)a_n$ for $n\geq0$Show that $\{a_n\}$ converges and find its limit..

How to find this? I got lost when i tried second order difference equation...

Thanks a lot!