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arithmetic progression of maximum length l

Source: Moldova TST 1998

August 8, 2023
arithmetic sequence

Problem Statement

Let M={1n∣n∈N}M=\{\frac{1}{n}|n\in\mathbb{N}\}. Numbers a1,a2,…,ala_1,a_2,\ldots,a_l from an arithmetic progression of maximum length ll (l≥3)(l\geq 3) if they verify the properties: a) numbers a1,a2,…,ala_1,a_2,\ldots,a_l from a finite arithmetic progression; b) there is no number b∈Mb\in M such that numbers b,a1,a2,…,alb,a_1,a_2,\ldots,a_l or a1,a2,…,al,ba_1,a_2,\ldots,a_l, b form a finite arithmetic progression. For example numbers 16,13,12∈M\frac{1}{6},\frac{1}{3},\frac{1}{2}\in M form an arithmetic progression of maximum length 33. a) FInd an arithmetic progression of maximum length 19981998. b) Prove that there exist maximum arithmetic progressions of any length l≥3l \geq 3.
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