Convergence and the limit of the sequence $x_n=\frac {a_n}{b_n}$ where $(1+\sqrt3)^n=a_n+b_n\sqrt3, n \ge 1$ with $a_n,b_n$ integers

Question

I have written the binomial expansion of $(1+\sqrt3)^n$ as

$$\binom n0 1 + \binom n1 \sqrt3 + \binom n2 3 +\binom n3 3\sqrt3 + ...$$

but I couldn't make any use of it. Can you give me a starting point to solve the problem?

Thanks for your effort and time in advance.

Answer 1

It is not hard to see that $$(1-\sqrt3)^n=a_n-b_n\sqrt3$$

Now solve the system for $a_n$ and $b_n$...