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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?

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    If you deduce that $$t_{n+5}=t_n$$is equivalent to the original equation then you should see that the general term would be written in terms of the fifth roots of unity. It's much easier to just use the pattern as suggested. – Peter Foreman Jun 27 '20 at 16:45
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    Several symbolic recurrence solvers I checked didn't return anything. Of course, this doesn't mean there isn't another way to proceed, but I think just computing 6-7 terms and realizing it's periodic is the best practical approach. – Integrand Jun 27 '20 at 17:22
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    Using the substitution $s_n=5t_n$, the recurrence relation reduces to $$s_n=\frac{s_{n-1}+1}{s_{n-2}}.$$ According to this posting, its periodicity is related to a non-trivial and deep topic called Laurent phenomenon algebra. – Sangchul Lee Jun 27 '20 at 17:50

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