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## 'Farey Fibonacci' printed from http://nrich.maths.org/

Denoting the Fibonacci numbers $1,1,2,3,5,8,...$ by $f_n$ where
$f_n=f_{n-1}+ f_{n-2}$ prove for all positive integer values of $n$
that $\frac{f_n}{f_{n+2}}$ and $\frac{f_{n+1}}{f_{n+3}}$ are Farey
neighbours, that is $|f_{n+1}f_{n+2}-f_nf_{n+3}|=1$.

Show that the mediant of $\frac{f_n}{f_{n+2}}$ and
$\frac{f_{n+1}}{f_{n+3}}$ is $\frac{f_{n+2}}{f_{n+4}}$.

See the problem

Farey Neighbours.