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`(naturally,_{n}`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`closest to_{n}`x`, in the order in which new terms appear with increasing`n`.