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.


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