Problem:
Define a sequence recursively by t1=20,t2=21, and
tn=25tn−25tn−1+1
for all n≥3. Then t2020 can be written as qp, where p and q are relatively prime positive integers. Find p+q.
Solution:
Define a new sequence sn=5tn so that the recurrence becomes
sn=sn−2sn−1+1.
We compute the initial terms:
s1=100, s2=105
Then:
s3=100105+1=5053
s4=1055053+1=5250103
s5=50535250103+1=52505353⋅5350=105101
s6=5250103105101+1=100, and
s7=105101100+1=105.
So the sequence is periodic with period 5. Therefore,
s2020=s5=105101 and t2020=5s2020=525101.
The requested sum is 101+525=626.
The problems on this page are the property of the MAA's American Mathematics Competitions