Just a few moments ago, I have asked a question regarding a proof that my math teacher asked me to solve. But it turned out to be a joke; he was asking me to solve the ABC conjecture, which apparently is a known open question in the math community. Now I have another extra credit proof(my teacher gave 5 of them; hopefully they aren't all just jokes), but I'm having trouble figuring out if this is just another joke question or not. It is as follows:
Consider a sequence $(a_n)$ that has an initial value of $a_0=k$, where $k$ is a positive integer. Every term after that is then defined recursively as follows:$$a_n=\begin{cases}\frac{a_{n-1}}2&\text{if }a_{n-1}\equiv0\pmod2\\3a_{n-1}+1&\text{if }a_{n-1}\equiv1\pmod2\end{cases}$$Prove or disprove the statement that there are infinitely many values for $k$ such that $(a_n)$ will reach $1$ at some point.
My initial thought was to try out small values for $k$. I first tested $k=3$, which produces the following sequence:$$3,10,5,16,8,4,2,1$$ I noticed that if the sequence reaches any power of $2$, then it will eventually reach $1$. But after this, I don't know what I can do. I suppose I can try to find an explicit formula for this recursive sequence but I feel like that would be a hard task.
This problem is pretty easy to understand, yet the proof still eludes me. I didn't spend too much time on this, thinking it might be another joke, but I'm not sure if that's the case here. Any thoughts?
People aren't reading the question carefully. This is NOT the Collatz conjecture, but a different problem about the Collatz sequence. The question given is easy to solve: Consider the infinite set $\{ 2^n | n = 1, 2, 3, \ldots\}.$