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Farey Neighbours

Age 16 to 18 Challenge Level:

Start by constructing some Farey Sequences.

Find out more about the mediant of two fractions in Mediant Madness.

If you haven't yet met Proof by Induction, you may like to read this article.

For your proof by induction, your inductive hypothesis could be that if $\frac bd$ and $\frac ac$ are Farey Neighbours with $c,d \leq n$, then $ad-bc=1$.
Then consider which fractions can be Farey Neighbours of $\frac bd$ and $\frac ac$ in $F_{n+1}$.