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?

Not Continued Fractions

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

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 $f_{2000}(2000)$

Number theory. Recurrence relations. Creating and manipulating expressions and formulae. Inequalities. Calculating with fractions. Generalising. Mathematical reasoning & proof. Making and proving conjectures. Polynomial functions and their roots. Working systematically.

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There's a Limit

Not Continued Fractions

Comparing Continued Fractions

And So on - and on -and On

Age 16 to 18 Challenge Level: