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t/a_t = k if r / a_r = k + 1

Source: New Zealand NZMOC Camp Selection Problems 2016 p8

September 19, 2021
algebraSequence

Problem Statement

Two positive integers rr and kk are given as is an infinite sequence of positive integers a1≤a2≤a3≤..a_1 \le a_2 \le a_3 \le .. such that rar=k+1\frac{r}{a_r}= k + 1. Prove that there is a positive integer tt such that tat=k\frac{t}{a_t}= k.
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