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# Not Continued Fractions

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Age 14 to 18

Challenge Level

- Find all positive integers $x$, $y$ and $z$ such that: $$x +\cfrac{1}{y + \cfrac{1}{z}} = N = \frac{10}{7}$$
- Show that when $N=10/7$ is replaced by $N=8/5$ it is impossible to find positive integer values of $x$, $y$ and $z$ for which the finite continued fraction on the left hand side is equal to $N$. Find another fraction (rational number) $N$ for which the same is true.

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

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?