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Farey Sequences

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

If I gave you a list of decimals, you might find it quite straightforward to put them in order of size. But what about ordering fractions?

A man called John Farey investigated sequences of fractions in order of size - they are called Farey Sequences.

The third Farey Sequence, $F_3$, looks like this:
Third Farey Sequence
It lists in order all the fractions between $0$ and $1$, in their simplest forms, with denominators up to and including $3$.

Here is $F_4$:
Fourth Farey Sequence
Write down $F_5$.

Which extra fractions are in $F_5$ which weren't in $F_4$?

Use $F_5$ to help you complete $F_6$ and $F_7$.

Here are some questions to consider:

There are lots of extra fractions in $F_{11}$ which are not in $F_{10}$.
There are only a few extra fractions in $F_{12}$ which are not in $F_{11}$.
Can you explain why this is the case?
When will you need lots of extra fractions to get the next Farey Sequence?

Will every Farey Sequence be longer than the one before? How do you know?

So far, all the Farey Sequences have contained an odd number of fractions. Can you find a Farey Sequence with an even number of fractions?

In $F_4$, $\frac{3}{4}$ slotted in between $\frac{2}{3}$ and $\frac{1}{1}$. What do you notice about the fractions on either side when you slot in a new fraction?

Choose any three consecutive fractions from a Farey Sequence. Can you find a way to combine the two outer fractions to make the middle one?

To see how these sequences relate to some beautiful mathematical patterns see the pictures in the problem Ford Circles. There is no need to attempt this problem at this stage - it is aimed at older students.