i was solving problems about sequences from a book (it is not english).here is the problem.
suppose that $a_n=a_{n-1}^2-2, a_0=3$ n then find
$$\lim_{n\to\infty} (\prod_{n=0}^{N-1} a_n)^{\frac{1}{2N} }$$
in teh offcial solution it says , suppose $\alpha=\frac{3+ \sqrt{5} }{2}$ then $a_0=\alpha+{\alpha}^{-1}$ then using induction one can obtain:
$$a_n={\alpha}^{2^{n}}+{\alpha}^{-(2^{n})}$$ so it is pretty easy to se the $limit$ is $\alpha$.
the thing is , the solution is not clear for me, how we should find $\alpha$? or how we should solve such recurrence equation? thanks.