Farey sequences have some very elegant properties, that even turn out to be useful. All references to terms of Farey sequences here refer to the reduced form of the fractions.

- The difference between adjacent terms of a Farey sequence with denominators
`m` and `n` is precisely 1/`mn`.
- Suppose
`a`/`b` and `c`/`d` are *adjacent* terms of the Farey sequence `F`_{n} (naturally, `a,b` < `n`). Then the two terms remain adjacent in *all* Farey sequences up to `F`_{b+d}, at which point (`a`+`c`) / (`b`+`d`) appears between them (and in particular, (`a`+`c`) / (`b`+`d`) is in reduced form!)
- Let 0<
`x`<1 be a real number. The convergents of the continued fraction for `x` are precisely the terms of Farey sequences `F`_{n} closest to `x`, in the order in which new terms appear with increasing `n`.