Define a sequence recursively by $t_1 = 20, t_2=21$ and $$t_n = \frac{5t_{n-1} + 1}{25t_{n-2}}$$ for all $n \geqslant 3.$ Then $t_{2020}$ can be written as $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
This was asked on the AIME 2020 and while it can be solved by computing consecutive terms and finding the pattern that repeats every $5$th step I would like to know if this could be solved by finding the characterstic equation for $t_n$ or by some other alternative approach?