This is the fourth Farey sequence


F4 = 0
1
 , 1
4
 , 1
3
 , 1
2
 , 2
3
 , 3
4
 , 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:


F5 = 0
1
 , 1
5
 , 1
4
 , 1
3
 , 2
5
 , 1
2
 , 3
5
 , 2
3
 , 3
4
 , 4
5
 , 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 1/2 and 1/3 is 5/6 but the mediant (which you get by adding the numerators and denominators) of 1/2 and 1/3 is 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.


F4 = 0
1
 , ì
í
î 		0 + 1
1+4
 ü
ý
þ 		, 1
4
 , 1
3
 , ì
í
î 		1+1
3+2
 ü
ý
þ 		, 1
2
 , ì
í
î 		1+2
2+3
 ü
ý
þ 		, 2
3
 , 3
4
 , ì
í
î 		3+1
4+1
 ü
ý
þ 		, 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?