Difference between Adjacent Terms of Farey Sequence

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Theroem

Let $F_n$ be a Farey sequence of order $n$.

Let $\dfrac a b$ and $\dfrac c d$ be consecutive terms of $F_n$ such that $\dfrac a b < \dfrac c d$.

Then:

$\dfrac c d - \dfrac a b = \dfrac 1 {b d}$

or equivalently:

$b c - a d = 1$


Proof


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Examples

Arbitrary Example

Consider the Farey sequence $F_5$ of order $5$:

$\dfrac 0 1, \dfrac 1 5, \dfrac 1 4, \dfrac 1 3, \dfrac 2 5, \dfrac 1 2, \dfrac 3 5, \dfrac 2 3, \dfrac 3 4, \dfrac 4 5, \dfrac 1 1$

Consider the consecutive terms $\dfrac 3 5$ and $\dfrac 2 3$.

We have:

\(\ds \dfrac 2 3 - \dfrac 3 5\) \(=\) \(\ds \dfrac {2 \times 5 - 3 \times 3} {3 \times 5}\)
\(\ds \) \(=\) \(\ds \dfrac {10 - 9} {15}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {15}\)

$\blacksquare$


Sources

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