I'm not a math major, and I have very little experience with any proof. I've never written one. But I would first think this is a 3 part problem. One of n=even, n=odd and n=2^x. Proving all will become 2^x seems to be the goal.
Any even integer n divided by 2 is an integer half the value
Any odd integer n multiplied by 3 is an odd integer.
Any odd integer n plus one is an even integer
Through induction, any power of two in this sequence will end in 1. That is to say if n_0 is 2^x, then the sequence will always be even and always end in 1. 2/2 =1, 8/2/2/2=1.
As an infinite sequence, n will always become odd for any even n_0, not a power of 2.
Since odd n always have 1 added, they will eventually become a power of 2.