AMC10 2017 A

AMC10 2017 A · Q13

AMC10 2017 A · Q13. It mainly tests Remainders & modular arithmetic, Sequences in number theory (remainders patterns).

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 \(n \geq 2\). Thus the sequence starts 0, 1, 1, 2, 0, 2, … . What is \(F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}\)?

递归定义数列：\(F_0 = 0\)，\(F_1 = 1\)，对于所有 \(n \geq 2\)，\(F_n = F_{n-1} + F_{n-2}\) 除以 3 的余数。于是数列开始为 0, 1, 1, 2, 0, 2, … 。求 \(F_{2017} + F_{2018} + F_{2019} + F_{2020} + F_{2021} + F_{2022} + F_{2023} + F_{2024}\)？

(A) 6 6

(B) 7 7

(D) 9 9

(E) 10 10

Answer

Correct choice: (D)

正确答案：（D）

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 \(0 + 1 + 1 + 2 + 0 + 2 + 2 + 1 = 9\).

数列开始为 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2，… 注意模式重复，周期为 8。因此无论哪 8 个连续项相加，答案均为 \(0 + 1 + 1 + 2 + 0 + 2 + 2 + 1 = 9\)。

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