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'And So on - and on -and On' printed from https://nrich.maths.org/
The solution needs you to be systematic.
Start with $ f_{0} $, then work out $ f_{1}$, then work out $
f_{2}$
\begin{eqnarray}f_{0}(2000)&=& \frac{1}{1-2000}\\
&=& \frac{1}{-1999}\end{eqnarray}