Let f(x)=11−xf(x) = \frac{1}{1-x}f(x)=1−x1. Determine the expression for the nnn-th iterate fn(x)f_n(x)fn(x) where fn(x)=(f∘f∘⋯∘f)(x)f_n(x) = (f \circ f \circ \dots \circ f)(x)fn(x)=(f∘f∘⋯∘f)(x) (nnn times).
fn(x)=11−nxf_n(x) = \frac{1}{1-nx}fn(x)=1−nx1
fn(x)=x−1xf_n(x) = \frac{x-1}{x}fn(x)=xx−1 for n≡2(mod3)n \equiv 2 \pmod 3n≡2(mod3)
fn(x)=xf_n(x) = xfn(x)=x if nnn is a multiple of 3
fn(x)=11−xnf_n(x) = \frac{1}{1-x^n}fn(x)=1−xn1