6.31. In the Farey sequence $F_{n}$, compute the fractions adjacent to $1 / 2$.

78 Chapter 6. Rational and Irrational Numbers
6.31. Answer: $\frac{1}{2} \pm \frac{1}{2 n}$ for odd $n$; $\frac{1}{2} \pm \frac{1}{2 n-2}$ for even $n$.

For odd $n$, the numerator and denominator of the fraction $\frac{n \pm 1}{2 n}$ can be reduced by 2; for even $n$, the same is true for the fraction $\frac{n-1 \pm 1}{2 n-2}$. Therefore, the fractions specified in the answer are part of the Farey sequence. If $\frac{a}{b} < \frac{c}{d}$ are adjacent fractions in the Farey sequence, then $\frac{c}{d} - \frac{a}{b} = \frac{bc - ad}{bd} = \frac{1}{bd}$. If $b=2$ and $d \leqslant n$, then $\frac{1}{bd} \geqslant \frac{1}{2 n}$. For odd $n$, the proof is complete. For even $n$, it is sufficient to show that if $\frac{a}{b} = \frac{1}{2 n}$, then $d \neq n$. But if $a=1, b=2$, and $d=n$, then $1 = bc - ad = 2c - n$, so $n$ is odd.