- Determine the smallest gap in the sequence, and find two fractions that have this gap between them.
- The gap between the first two fractions is 1/20 – 0/1 = 1/20, as is the gap between the last two fractions. Not counting these fractions, find two fractions that have a gap that is as large as possible.
E3. A Fraction Sequence
Let S be the sequence of all proper fractions with denominator ≤ 20, arranged in order from smallest to largest, so the first fraction is 0/1 = 0 and the last one is 1/1 = 1. Represent each fraction in lowest terms. Define the gap between two consecutive fractions in this sequence to be the difference of the smaller from the larger.