I came across this problem in a Quant finance prep test book. This looks a bit like a geometric series, a bit like an arithmetic series, and a bit like a recursive function.
$ f = \frac{1}{1 + \frac{1} {1 + ...}} $
Which I would rewrite as, $ f = \frac{1}{1 + f} $ , however, this might not be valid.
How do I solve this problem? And what is the name of this sort of problem?
It is called a continued fraction, and this one, in particular, is equal to $\frac{\sqrt{5}-1}{2}$, which we get via the substitution you mentioned.
In order to show this is valid, we can simply show via induction that if we make a fraction of this form k fractions deep (e.g. if k=1 then the fraction would be 1/1, if k=2 then it would be 1/(1+1/1)), etc.), and this fraction is limited between two bounds, then the next fraction would also need to be in between those two bounds. In this case, we can show that a=1/(1+1/(1+...)) with depth k is between 0 and 2, then 1/(1+1/(1+...))=1/(1+a) with depth k+1 is between 1/(1+2)=1/3 and 1/(1+0)=1, and since 1/1 is between 0 and 2 the base case is good and we are done.