If $a_n+b_n\sqrt{3}=(2+\sqrt{3})^n$, then what's $\lim_{n\rightarrow\infty}\frac{a_n}{b_n}$? Let $a_n$ and $b_n$ be integers defined in the following way: $$a_n+b_n\sqrt{3}=(2+\sqrt{3})^n.$$ Compute $$\lim_{n\rightarrow\infty}\frac{a_n}{b_n}.$$
I tried expanding using binomial theorem:
$$ (2+\sqrt{3})^n=\binom{n}{0}2+\binom{n}{1}2^2\sqrt{3}+\binom{n}{2}2^3\cdot 3+\ldots+\binom{n}{n-1}2\sqrt{3}^{n-1}+\binom{n}{n}\sqrt{3}^{n}$$
and I think we have that: $$ a_n=\binom{n}{0}2+\binom{n}{2}2^3\cdot 3+\ldots+\binom{n}{n}\sqrt{3}^{n}$$
$$ b_n=\binom{n}{1}2^2\sqrt{3}+\ldots+\binom{n}{n-1}2\sqrt{3}^{n-1}$$
But, frankly, I do not know what to do next. 

Note that
$$a_n - b_n\sqrt{3} = (2-\sqrt{3})^n$$
Hence,
$$2a_n = (a_n - b_n\sqrt{3}) + (a_n + b_n\sqrt{3}) \Rightarrow a_n = \dfrac{(2+\sqrt{3})^n + (2-\sqrt{3})^n}{2}$$
Expressing $b_n$ we obtain:
$$b_n = \dfrac{(2+\sqrt{3})^n - a_n}{\sqrt{3}} = \dfrac{(2+\sqrt{3})^n - (2-\sqrt{3})^n}{2\sqrt{3}}$$
Now it should be easy to compute the limit of $\dfrac{a_n}{b_n}$. Since $(2-\sqrt{3})^n \to 0$, we can conclude that
$$\lim\limits_{n\to+\infty}\dfrac{a_n}{b_n} = \lim\limits_{n\to+\infty} \dfrac{(2+\sqrt{3})^n}{(2+\sqrt{3})^n/\sqrt{3}} = \sqrt{3}$$
