<s>[INST] 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. 
 [/INST] Hint: First observe that $a_n -b_n\sqrt{3}=(2-\sqrt{3})^n$. Combine with the assumption, we get $a_n^2 - 3b_n^2 =1$ for all $n$. Then $\frac{a_n^2}{b_n^2}-3=\frac{1}{b_n^2}$. You should prove that $b_n \to \infty$. Hence, $\frac{a_n}{b_n} \to \sqrt{3}$ since $\frac{a_n}{b_n}>0$.
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