Difference between Adjacent Terms of Farey Sequence/Examples

Examples of Difference between Adjacent Terms of Farey Sequence

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:

\(\ds \dfrac 2 3 - \dfrac 3 5\)

\(=\)

\(\ds \dfrac {2 \times 5 - 3 \times 3} {3 \times 5}\)

\(\ds \)

\(=\)

\(\ds \dfrac {10 - 9} {15}\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {15}\)

$\blacksquare$