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There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

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Comparing Continued Fractions

Which of these continued fractions is bigger and why?

And So on - and on -and On

Age 16 to 18 Challenge Level:
Let $$ f_{0}(x)= \frac{1}{1-x}$$ and $$f_{n}(x)=f_{0}(f_{n-1}(x))$$ where $n = 1,2, 3, 4, ...$

Evaluate $f_{2000}(2000)$