This is the fourth Farey sequence

F_{4} = \frac{0}{1}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{1}{1}

which is the list, written in increasing order, of all the fractions with values between 0 and 1 that use only the numbers 0,1, 2, 3, 4 as numerators and denominators.

Now if we use the number 5 as well we get the fifth Farey sequence:

F_{5} = \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1}

Write down all the Farey sequences F ₁ ,F ₂ ,...,F ₁₀ .

Sometimes people make a mistake when adding fractions and, instead of finding the sum, they find the mediant. For example the sum of $\frac{1}{2}$ and $\frac{1}{3}$ is $\frac{5}{6}$ but the mediant (which you get by adding the numerators and denominators) of $\frac{1}{2}$ and $\frac{1}{3}$ is $\frac{2}{5}$ . If you add two positive fractions the sum is bigger than both of them but the mediant lies between them. .

Here is the fourth Farey with some mediants written in curly braces in the list.

F_{4} = \frac{0}{1}, {\frac{0 + 1}{1 + 4}}, \frac{1}{4}, \frac{1}{3}, {\frac{1 + 1}{3 + 2}}, \frac{1}{2}, {\frac{1 + 2}{2 + 3}}, \frac{2}{3}, \frac{3}{4}, {\frac{3 + 1}{4 + 1}}, \frac{1}{1}

What do you notice? How do the fourth and fifth Farey sequences relate to each other? Does the same relationship hold between the other Farey sequences that you have found?