Finding a specific term in a repeating sequence. Let $f(x) = \frac1{1-x}$,
and define the function $f^r$ by
$$f^r(x):=\underbrace{f(f(f(...f(f}_{r\text{ times}}(x))))).$$
I am asked to find to find $f^{653} (56)$.
I know that there are only $3$ different values which $f^{653}$ could be: $-1/55$, $55/56$ or $56$. Using a really tedious method, I got that the $653$-th term is $56$. The method I used took me about $20$ minutes and I am wondering if there is a "shortcut" approach to a problem like this.

Notice that $f^{2}(x) = \frac{x-1}{x}$ and that $f^{3}(x) = x$. Then reduce 653 mod 3. This gives $(653)\equiv 2 ~mod ~3$, so $f^{653}(x) =f^{2}(x)$.
