Question 164
For every integer $n \in \mathbb{N}^{*}$, denote by $v_{3}(n)$ the 3-adic valuation of $n$, that is, the largest integer $k$ such that $n$ is divisible by $3^{k}$. Put $u_{1}=2$ and $u_{n}=4 v_{3}(n)+2-\frac{2}{u_{n-1}}$ for all $n \geqslant 2$ (provided that $u_{n-1}$ is defined and nonzero).
Show that, for every strictly positive rational number $q$, there exists one and only one integer $n \geqslant 1$ such that $u_{n}=q$.
Show that, for every strictly positive rational number $q$, there exists one and only one integer $n \geqslant 1$ such that $u_{n}=q$.