Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Find the answer

4. Consider the sequence $x_{n}=(1+\sqrt{2}+\sqrt{3})^{n}$ . Let
$x_{n}=a_{n}+b_{n} \sqrt{2}+c_{n} \sqrt{3}+d_{n} \sqrt{6} \text {, }$

where, $a_{n}, b_{n}, c_{n}, d_{n} \in \mathbf{Z}_{+}$ .
Find $\lim _{n \rightarrow+\infty} \frac{b_{n}}{a_{n}}$ , $\lim _{n \rightarrow+\infty} \frac{c_{n}}{a_{n}}$ , and $\lim _{n \rightarrow+\infty} \frac{d_{n}}{a_{n}}$ .

Solution

The answer is:
$\lim _{n \rightarrow+\infty} \frac{b_{n}}{a_{n}}=\frac{\sqrt{2}}{2}, \lim _{n \rightarrow+\infty} \frac{c_{n}}{a_{n}}=\frac{\sqrt{3}}{3}, \lim _{n \rightarrow+\infty} \frac{d_{n}}{a_{n}}=\frac{\sqrt{6}}{6} .$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.