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Rationals Between...

What fractions can you find between the square roots of 65 and 67?

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?

Harmonic Triangle

Age 14 to 16
Challenge Level

$$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$$
$$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$$
$$\frac{1}{12} = \frac{1}{20} + \frac{1}{30}$$

Look at the diagonal lines running from the right down to the left. The fractions in the first one are $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and so on.

If $\frac{1}{n}$ is at the end of the nth row, the fraction above it must be $\frac{1}{n-1}$ and the fraction below it must be $\frac{1}{n+1}$.

Have a look at the second diagonal (the one formed by taking the second number in each row: it starts $\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}$.

Can you find a pattern for these numbers so that you can work them out easily (without having to subtract fractions)?

Can you explain why the pattern works?