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Épreuve de section européenne

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Baccalauréat, toutes séries Session de juin 2012

Épreuve de section européenne

Farey sequence

In mathematics, the Farey sequence of ordernis the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal ton, arranged in order of increasing size.

Each Farey sequence starts with the value 0, denoted by the fraction 01, and ends with the value 1, denoted by the fraction 11 (although some authors omit these terms).

For example, the Farey sequence of order 4 is denotedF4 and is F4 =

0

1,1 4,1

3,1 2,2

3,3 4,1

1

There are many interesting things about Farey sequences but we will focus on how to build Fn, the Farey sequence of order n, whenn is a given natural number. Here is a way to do it :

The Farey sequence of orderncontains all of the terms of the Farey sequences of lower orders.

In particular Fn contains all of the terms of Fn−1. So knowing Fn−1, to get Fn, you only have to add the irreducible fractions between 0 and 1 whose denominator is n.

For example, F6 consists of F5 together with the fractions 16 and 56.

From various sources

Questions

1. Using the definition, give the Farey sequence of order 3.

2. Using the method explained above, deduce F5 fromF4.

3. Explain why the Farey sequence of order ncontains all of the terms of the Farey sequence of order n−1.

4. Another method to get Fn from Fn−1 is to use the following property : If ab, pq and cd are three consecutive terms in a Farey sequence such that ab < pq < cd then pq = ab++dc. Illustrate this property with the values from F4,

5. Admitting that ab and cd are two consecutive terms in a Farey sequence if and only if bc−ad= 1, prove the property given in the previous question.

2012-15 – Farey sequence

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