The trick to recognizing these Farey sequences is to first look at the denominators.
For #F_3#, the denominators (in order) are:
#{1,3,2,3,1}#
The number 3 is the biggest denominator and there are 5 terms.
For #F_4#, the denominators are:
#{1,4,3,2,3,4,1}#
The number 4 is the biggest denominator and there are 7 terms.
The pattern for the denominators of all subsequent Farey sequences should be obvious.
The number of terms is always increased by 2. So, for #F_5#, the number of terms will be 9. And for #F_6#, the number of terms will be 11.
The first term is always #0//1#. The last term is always #1//1#. The middle term is always #1//2#. After 0, the numerators up to #1//2# will always be #1#. The numerators after #1//2# will simply count from 2 to the max number.
It may be easier to see if you let the last term #1/1# be equal to the maximum denominator, divided by itself. So,
Last term of #F_3# is #3//3#
Last term of #F_4# is #4//4#
Last term of #F_5# is #5//5#, etc.
There is always two more terms shoved in each subsequent Farey sequence. Let #m# be the max denominator. Always, just after the zero term (the first term in the sequence) will be #1//m#. And finally, right before the final 1 term (the next to last in the sequence) will be #(m-1)//m#.
I'll let you explore the "questions to consider" on your own!
