Given the functions $f_0,...,f_n$ with $f_0(x)=\frac{1}{x-2}$ and $f_{k+1}(x)=\frac{1}{1-f_k(x)}$ for every $k=0,1,...,n-1$. Evaluate $f_{2000}(2001)$

Question

Given the functions $f_0,f_1,...,f_n$ with $f_0(x)=\frac{1}{x-2}$ and $f_{k+1}(x)=\frac{1}{1-f_k(x)}$ for every $k=0,1,...,n-1$. Evaluate $f_{2000}(2001)$.

I'm trying to work out this question by attempting to find a pattern, but I can't seem to find one. Based on a pattern which I see, I believe that the number is negative, but I can't work out the question. Could you please explain to me how to solve it?

Answer 1

Hint:

$$f_1(x)=\frac{1}{1-\frac{1}{x-2}}=\frac{x-2}{x-3}$$ $$f_2(x)=\frac{1}{1-\frac{x-2}{x-3}}=3-x$$ $$f_3(x)=\frac{1}{1-(3-x)}=\frac{1}{x-2}$$