Question 10
Let $h$ be a positive integer and let $a_{n}$ the sequence defined for recurrence as follows: $$
\begin{aligned}
a_{0} & =1 \\
a_{n+1} & = \begin{cases}\frac{a_{n}}{2} & \text{ if } a_{n} \text{ is even } \\
a_{n}+h & \text{ if } a_{n} \text{ It's odd.}\end{cases}
\end{aligned}
$$ (for example, if $h=27$ we have: $a_{1}=28, a_{2}=14, a_{3}=7, a_{4}=34, a_{5}=17, a_{6}=44, \ldots$) For what values of $h$ does $n\gt 0$ exist for which $a_{n}=1$?
\begin{aligned}
a_{0} & =1 \\
a_{n+1} & = \begin{cases}\frac{a_{n}}{2} & \text{ if } a_{n} \text{ is even } \\
a_{n}+h & \text{ if } a_{n} \text{ It's odd.}\end{cases}
\end{aligned}
$$ (for example, if $h=27$ we have: $a_{1}=28, a_{2}=14, a_{3}=7, a_{4}=34, a_{5}=17, a_{6}=44, \ldots$) For what values of $h$ does $n\gt 0$ exist for which $a_{n}=1$?