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Consider the sequence $(a_k)_{k\geq 1}$ of positive rational numbers defined by $a_1 = \frac{2020}{2021}$ and for $k \geq 1,$ 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 $t.$

 Mar 10, 2021
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The sum of all positive integers t is 20 + 40 + 60 = 120.

 Mar 10, 2021

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