I recently run into a question that asked me to find a number $n$ if for $k$ times, $n$ has been halved and subtracted $0.5$ and, at the end, $n$ becomes $0$. I don't know if this is of any relevance but the question was about people in a bus and in every stop, half of the people plus half a person would leave the bus until the bus is empty.
Given $k$, the answer was just $2^{k-1}$, I don't quite understand how to get there, the only thing I got was:
$$\left(\sum_{i=1}^{k}\frac{n+1-2^{i-1}}{i^2}\right)+\frac{k}{2}=n$$
I would appreciate any help.
P.D.: I'm positive this is very basic to almost all of you, but I'm trying to get started in the mathematical world as I've found it to be extremely interesting.