Question 165
For a natural number $n$, consider the sequence $(a_k)_{k=1}^{\infty}$ given by $a_1 = \frac{1}{n}$ and the recurrent relation
$$
a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor,
$$
for all $k \ge 1$. Determine all the values of $n$ for which the sequence is eventually constant.
$$
a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor,
$$
for all $k \ge 1$. Determine all the values of $n$ for which the sequence is eventually constant.