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Fibonacci Mod Three?

Source: 2017 AMC10A #13

February 8, 2017
AMCAMC 10AMC 10 A2017 AMC 10Arecursion

Problem Statement

Define a sequence recursively by F0=0F_0 = 0, F1=1F_1 = 1, and Fn=F_n = the remainder when Fn−1+Fn−2F_{n-1} + F_{n-2} is divided by 33, for all n≥2n \ge 2. Thus the sequence starts 0,1,1,2,0,2…0,1,1,2,0,2 \ldots. What is F2017+F2018+F2019+F2020+F2021+F2022+F2023+F2024F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 7<spanclass=′latex−bold′>(C)</span> 8<spanclass=′latex−bold′>(D)</span> 9<spanclass=′latex−bold′>(E)</span> 10<span class='latex-bold'>(A)</span>\ 6\qquad<span class='latex-bold'>(B)</span>\ 7\qquad<span class='latex-bold'>(C)</span>\ 8\qquad<span class='latex-bold'>(D)</span>\ 9\qquad<span class='latex-bold'>(E)</span>\ 10
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