2017年AMC10竞赛A第13题
2017 AMC10A P13  Define a sequence recursively by $$F_0 =0$$, $$F_1=1$$, and $$F_n=$$the remainder when $$F_{n-1}+ F_{n-2}$$ is divided by $$3$$, for all $$n≥ 2$$. Thus the sequence starts $$0$$, $$1$$, $$1$$, $$2$$, $$0$$, $$2$$, $$\cdots$$ What is $$F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}$$?  用$F_0=0$$，$$F1=1$$，以及Fn=F_{n-1}+ F_{n-2}$除以$3$$时的余数来递归地定义一个数列(n≥2)。$$F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}$$?=?
美国AMC10/12->Knowledge Point->Combination->Reasoning->Recurrence and Recursion
A patten starts to emerge as the function is continued. The repeating patten is $$0$$, $$1$$, $$1$$, $$2$$, $$0$$, $$2$$, $$2$$, $$1$$ $$\ldots$$. The problem asks for the sum of eight consecutive terms in the sequence.  Because there are eight numbers in the repeating sequence, we just need to find the sum of the numbers in the sequence, which is $$(\rm D) 9$$. 