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Do Unto Caesar   Farey Sequences   This problem concerns a
sequence of numbers named after John Farey, an English
geologist. He wrote about these sequences in an article called
"On a curious property of vulgar fractions", published in 1816.
Back in those days, "vulgar fractions" were what we now call
"improper fractions"; "vulgar" meant (and still does mean)
commonplace, rather than obscene or uncouth! The "curious
property" that Farey noticed is something that you are asked to
explore in this problem, so more on this later...  The first
task was to work out $F_5$. Kieran, from Oakwood Park Grammar
School worked out the next sequence by starting with the
previous one, using it as a framework. From this, he then
slotted in the new fractions, rather than creating each new
sequence from scratch.  Weida, from Collaton St Mary Primary
School, submitted the correct sequence for $F_5$:  F5 =
$\frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}