Number of pieces produced by cutting a folded ribbon A piece of ribbon 1 metre long is folded in half so that the two ends are on top of each other. This doubled piece of ribbon is then folded in half again. The folded piece of ribbon is then cut  right through its midway point (see my drawing below).

This produces 5 pieces of ribbon of lengths: $\frac{1}{8},\frac{1}{4},\frac{1}{4}, \frac{1}{4}, \frac{1}{8}$.
My question is what happens after $n$ folds?
Trying a few more examples has led me to guess that after $n$ folds: there are $2^n + 1$ pieces of ribbon, with two strands of length $\frac{1}{2^{n+1}}$ and the rest of length $\frac{1}{2^{n}}$. I am having difficulty creating a convincing proof of this.
I came across this problem on an Oxford University Thinking Skills Admissions test ( Question 38, page 23 ). If this is a well known problem, I would also be interested in a source to read about the question in more detail.

HINT: I think you can convince yourself with this argument:

*

*After $N$ folds, how many layers "thick" is the ribbon?


*The cut will create $2$ open ends per layer.  So how many total open ends are there after the cut?  Don't forget the $2$ original open ends (the ends of the original ribbon).


*No. of pieces is related to total no. of open ends is a very simple way.
