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Farey Fibonacci

Age 16 to 18 Short Challenge Level:

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.