Problem:
Define a sequence recursively by F0​=0,F1​=1, and Fn​= the remainder when Fn−1​+Fn−2​ is divided by 3, for all n≥2. Thus the sequence starts 0,1,1,2,0,2,…. What is F2017​+F2018​+F2019​+ F2020​+F2021​+F2022​+F2023​+F2024​?
Answer Choices:
A. 6
B. 7
C. 8
D. 9
E. 10
Solution:
The sequence starts 0,1,1,2,0,2,2,1,0,1,1,2,… Notice that the pattern repeats and the period is 8. Thus no matter which 8 consecutive numbers are added, the answer will be 0+1+1+2+0+2+2+1=(D)9​.
The problems on this page are the property of the MAA's American Mathematics Competitions