MIE 2015/Day 2/Problem 2

Problem 2

Let the functions $f_n$, for $n\in\{0,1,2,3,...\}$, such that $f_0(x)=\frac{1}{1-x}$ and $f_n(x)=f_0(f_{n-1}(x))$, for every $n\geq1$.

Compute $f_{2016}(2016)$.


First, let see the case $n=1$




Now, when $n=2$




Now, when $n=3$


At this point it's easy to see the pattern. So, we just find the remainder of 2016 by 3.




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