# Difference between revisions of "Farey sequence"

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Where <math>F_n</math> denotes a Farey sequence of order <math>n</math>. | Where <math>F_n</math> denotes a Farey sequence of order <math>n</math>. | ||

+ | |||

+ | ==Proof Sketch== | ||

+ | Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+1}</math>? | ||

+ | Which is bigger, <math>\frac{a}{b}</math> or <math>\frac{a+1}{b+2}</math>? | ||

+ | Do you see a pattern? | ||

==Properties== | ==Properties== |

## Revision as of 04:12, 30 January 2021

A **Farey sequence** of order is the sequence of all completely reduced fractions between 0 and 1 where, when in lowest terms, each fraction has a denominator less than or equal to . Each fraction starts with 0, denoted by the fraction 0/1, and ends in 1, denoted by the fraction 1/1.

## Examples

Farey sequences of orders 1-4 are:

Where denotes a Farey sequence of order .

## Proof Sketch

Which is bigger, or ? Which is bigger, or ? Do you see a pattern?

## Properties

### Sequence length

A Farey sequence of any order contains all terms in a Farey sequence of lower order. More specifically, contains all the terms in . Also, contains an extra term for every number less than relatively prime to . Thus, we can write

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