Problem

Source: Israel National Olympiad 2018 Q2

Tags: arithmetic sequence, geometric sequence, number theory, algebra, geometry



An arithmetic sequence is an infinite sequence of the form $a_n=a_0+n\cdot d$ with $d\neq 0$. A geometric sequence is an infinite sequence of the form $b_n=b_0 \cdot q^n$ where $q\neq 1,0,-1$. Does every arithmetic sequence of integers have an infinite subsequence which is geometric? Does every arithmetic sequence of real numbers have an infinite subsequence which is geometric?