Difference between Adjacent Terms of Farey Sequence/Examples
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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:
\(\ds \dfrac 2 3 - \dfrac 3 5\) | \(=\) | \(\ds \dfrac {2 \times 5 - 3 \times 3} {3 \times 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {10 - 9} {15}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {15}\) |
$\blacksquare$