Question 56
$$
a_0 = 1, \quad a_{n+1} = \begin{cases} \frac{a_n}{2}, & \text{if } a_n \text{ is even,} \\ a_n + d, & \text{if } a_n \text{ is odd.} \end{cases}
$$
Determine all $d$ such that $a_n = 1$ for some $n \gt 0$. (Italy 2005)
a_0 = 1, \quad a_{n+1} = \begin{cases} \frac{a_n}{2}, & \text{if } a_n \text{ is even,} \\ a_n + d, & \text{if } a_n \text{ is odd.} \end{cases}
$$
Determine all $d$ such that $a_n = 1$ for some $n \gt 0$. (Italy 2005)