real analysis - Recursive relation in the square root form.

I apologize if this has been asked before, but I have been struggling to show the convergence and the limit value of the following recursive relation:

$x_{n+2}=\sqrt{x_n x_{n+1}}$ with initial values $x_1=a$ and $x_2=b$ for $0

I showed by induction that $ x_{2n-1}. I know that if I could show that $\lim_{n\to\infty} x_{2n}-x_{2n-1}=0$, then this implies convergence. However, I could not see the pattern. Moreover, I could not find the actual limit either. Thanks for any help!