# 2021 AIME I #10

I can tell when students are ramping up their preparation for AMC because my solution videos get a lot more views. This one has been receiving a lot of views lately.

Consider the sequence $(a_k)_{k \ge 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k \ge 1$, if $a_k = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then $a_{k+1} = \frac{m+18}{n+19}$. Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\frac{t}{t+1}$ for some positive integer $t$.