Competition problem — Algebra — Maths Olympiad Prep

Library / /242 of 520

Algebra Difficulty 5.5 AIME, harder Prove it

Let $s_{1}, s_{2}, \ldots$ be an infinite arithmetic progression of distinct positive integers. Prove that $s_{s_{1}}, s_{s_{2}}, \ldots$ is also an infinite arithmetic progression of distinct positive integers.

Solution

Let $s_{n}=a n+b$ for some integers $a$ and $b$ . Then $s_{s_{n}}=a(a n+b)+b=$ $a^{2} n+(a b+b)$ , which is also an arithmetic progression.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.