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arithmetic sequence whose inverses form also arithmetic sequence

Source: 2023 Belgium, VWO Flanders MO p1

March 25, 2024
arithmetic sequencealgebra

Problem Statement

An arithmetic sequence is a sequence of numbers for which the difference between two consecutive numbers applies terms is constant. So this is an arithmetic sequence with difference 56\frac56: 13,76,2,176,113,92.\frac13,\frac76, 2,\frac{17}{6},\frac{11}{3},\frac92. The sequence of seven natural numbers 6060, 7070, 8484, 105105, 140140, 210210, 420420 has the property that the sequence inverted numbers (i.e. the row 160\frac{1}{60}, 170\frac{1}{70}, 184\frac{1}{84}, 1105\frac{1}{105}, 1140\frac{1}{140}, 1210\frac{1}{210},1420\frac{1}{420} ) is an arithmetic sequence.
(a) Is there a sequence of eight different natural numbers whose inverse numbers are one form an arithmetic sequence?
(b) Is there an infinite sequence of distinct natural numbers whose inverses are form an arithmetic sequence?
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