Let $x_1 = 1, x_2 = 2 $, and for $n=1,2,...$ following recurrence is defined,
$$x_{2n+1} = \frac{5x_{2n-1} + 2x_{2n}}{7} , x_{2n+2} = \frac{2x_{2n-1} + 5x_{2n}}{7}$$
I need to show that $\lim_{n\to\infty} x_n$ exists and also find the lim.
My attempt: I tried to look at the difference between consecutive terms but it didn't worked, could anyone suggest any hint on how should I proceed...