Sequence of elements $x_i \in \bar{Q}$

Question

I am reading about applications of Galois theory to polynomials, but when using it for a degree 5 polynomial I got confused by the following: I need to show that any sequence of elements $x_i \in \bar{Q}$ (or in any field, for that matter) that satisfies the recursion

$x_{i-1}x_{i+1} = x_{i} +1$ for all $i\geq 1$ is necessary periodic of period 5. How do I show this?

Thanks

Answer 1

You can write $x_2,x_3,x_4,x_5,x_6$ as rational functions of $x_0$ and $x_1$. If you find that $x_5 = x_0$ and $x_6 = x_1$ then you will have proved that all those sequences are $5$-periodic.