A friend of mine gave me a math problem whose answer turned out to be
$$f(n) = 1 - (1 - f(n - 1) \times (1 - p)) ^ 2$$
for some fixed $p$.
I'm trying to find a closed-form solution to the recurrence, if this is possible... but no matter what I try to do (e.g. computing $f(n) - f(n - 1)$, computing $f'(n)$, etc.) I don't get anywhere.
How would I solve this recurrence relation?