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The $n$ 'th Farey sequence is the ascending sequence of all rationals $\{0 \le \frac{a}{b} \le 1 : b \le n\}$ .
The first 5 Farey sequences are
| 1 |
$ \frac{0}{1} < \frac{1}{1} $ |
| 2 |
$ \frac{0}{1} < \frac{1}{2} < \frac{1}{1} $ |
| 3 |
$ \frac{0}{1} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{1}{1} $ |
| 4 |
$ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1} $ |
| 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} $ |
Farey sequences are a singularly useful tool in understanding the convergents that appear in continued fractions. The convergents for any irrational $\alpha$ can be found: they are precisely the closest number to $\alpha$ on the sequences $F_n$ .
It is also of value to look at the sequences $F_n$ as $n$ grows. If $\frac{a}{b}$ and $\frac{c}{d}$ are reduced representations of adjacent terms in some Farey sequence $F_n$ (where $b,d\le n$ ), then they are adjacent fractions; their difference is the least possible: $$ \left|\frac{a}{b}-\frac{c}{d}\right| = \frac{1}{bd}. $$ Furthermore, the first fraction to appear between the two in a Farey sequence is $\frac{a+c}{b+d}$ , in sequence $F_{b+d}$ , and (as written here) this fraction is already reduced.
An alternate view of the ``dynamics'' of how Farey sequences develop is given by Stern-Brocot trees.
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