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,…. What is F_2017+F_2018+F_2019+F_2020+ F_2021+F_2022+F_2023+F_2024?

2017 AMC 10A Problem 13 — Solution

2017 AMC 10A Problem 13

Below is the professionally curated solution for Problem 13 of the 2017 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 10A solutions, or check the answer key.

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Concepts:Fibonacci modular arithmetic pattern recognition

Difficulty rating: 1140

13.

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 $0,1,1,2,0,2,\ldots.$ What is $F_{2017}+F_{2018}+F_{2019}+F_{2020}+$ $F_{2021}+F_{2022}+F_{2023}+F_{2024}?$

$6$

$7$

$8$

$9$

$10$

Solution:

Let us list out the first few values to see if we can find a pattern in this sequence.

$0, 1, 1, 2, 0, 2, 2, 1, 0, 1, \cdots$

From this we can see that the pattern repeats every $8$ terms.

The desired answer is the sum of $8$ consecutive numbers, which is fixed. This sum is $0 + 1 + 1 + 2 + 0 + 2$ $+ 2 + 1 = 9.$ Thus, D is the correct answer.

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