2020 AIME II Problem 6

Define a sequence recursively by t_1 = 20, t_2 = 21, andt_n = \frac{5t_{n-1}+1}{25t_{n-2}}for all n \ge 3. Then t_{2020} can be written as \frac{p}{q}, where p and q are relatively prime positive integers. Find p+q.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.