Define a sequence like below
$$\{x_n\}_{n=1}^{\infty}\\x_n=2+\frac{k_1}{2+\frac{k_2}{\ddots+\frac{k_n}{2}}}$$and
$$k_1,k_2,...,k_n \in \{5,20\}$$
What is the $max\{x_n\},min\{x_n\} ?$
I tried to put all of them $5,20$ but , get stuck . The key answer is $max=10, min=2.5$
can you help me . Thanks in advance.
2026-03-29 22:28:18.1774823298
Sequence of continued fractions
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For $a_i>0$ $$2+\frac{k_1}{a_1}\quad\text{is maximized when $k_1=20$ and $a_1$ is minimized}$$ $$a_1=2+\frac{k_2}{a_2}\quad\text{is minimized when $k_2=5$ and $a_2$ is maximized}$$ $$a_2=2+\frac{k_3}{a_3}\quad\text{is maximized when $k_3=20$ and $a_3$ is minimized}$$ and so on. Therefore, $\lim\sup_{n\to \infty}x_n$ is given by $$2+\frac{20}{2+\frac{5}{2+\frac{20}{2+\frac{5}{2+...}}}}$$ A similar argument works when you want to find the minimum.