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x_r - x_s = 2018 if x_n < x_{n+1} <=2n

Source: Mathematics Regional Olympiad of Mexico Northeast 2018 P4

September 12, 2022
algebraSequence

Problem Statement

We have an infinite sequence of integers {xn}\{x_n\}, such that x1=1x_1 = 1, and, for all n≥1n \ge 1, it holds that xn<xn+1≤2nx_n < x_{n+1} \le 2n. Prove that there are two terms of the sequence,xr x_r and xsx_s, such that xr−xs=2018x_r - x_s = 2018.
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