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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?

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Recurrence relations don't always have nice solutions. A famous example is the logistic map. You might be lucky though, I don't know. –  Ben Millwood Sep 2 '13 at 0:17
    
You might try some numerical experiments and plots to see if the sequence looks familiar. Perhaps some special values of $p$ might help. Usually you settle for some asymptotics instead of closed form solutions. –  Maesumi Sep 2 '13 at 0:29
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