MathNet 2020 — Math Olympiad Vault — Part 05 Question 127

Let $k$ be a positive integer greater than $1$ and let $(a_n)_{n \ge 1}$ be a sequence of pairwise distinct positive integers. Show that the set $M = \{a_1^k, a_2^k, a_3^k, \dots\}$ does not contain an infinite arithmetic sequence.