I have some hard time solving some of the following problems:
Consider the recurrence relation $x_n=2(x_{n-1}+x_{n-2})$
1) Show that the general solution is $z_n=\alpha(1+\sqrt3)^n+\beta (1-\sqrt3)^n$ and determine whether the difference equation is stable.
2) Show that starting values $x_1=1$ and $x_2=(1-\sqrt3)$ corresponds to the solution with $\alpha =0$ and $\beta=(1-\sqrt3)^{-1}$
3) For which values of $\alpha, \beta$ ∈ ℝ does the general solution converge? Determine the order of the rate of convergence for these values.
I don't know how to find the solution to the recurrence relation, the literature we have been given from out teacher only about convergent sequences (linear convergence, superlinear convergence, quadratic convergence) so i dont know how the heck they went from $x_n=2(x_{n-1}+x_{n-2})$ to $z_n=\alpha(1+\sqrt3)^n+\beta (1-\sqrt3)^n$. So I need someone who can push me in the right direction step by step.