I have found the general solution for a reccurence relation:
$$z_n=\alpha(1+\sqrt{3})^n+\beta(1-\sqrt{3})^n,$$
and is asked for which values of $\alpha$ and $\beta$ it converges. My first thought is that it converges for $\alpha=0$ and $\beta\in\mathbb{R}$, as $(1+\sqrt{3})^n\to\infty$ and $(1-\sqrt{3})^n\to0$, however, it seems too simple. Any thoughts on this would be greatly appreciated.
Best Answer
Suppose that $(z_n)$ is convergent, then the sequence $(z_n-\beta(1-\sqrt{3})^n)$ is convergent for each $ \beta$. But then $(\alpha(1+\sqrt{3})^n)$ is convergent. This is the case $ \iff \alpha =0$.
Result: $(z_n)$ converges $ \iff \alpha =0$ and $\beta$ is arbitrary.