Let L_1 = 1, L_2 = 3, and L_n+2 = L_n+1 + L_n for n ≥ 1. How many terms in the sequence L_1, L_2, L_3, …, L_2023 are even?

2023 AMC 10B Problem 6 — Solution

2023 AMC 10B Problem 6

Attempt Problem 6 of the 2023 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Let $L_1 = 1,$ $L_2 = 3,$ and $L_{n+2} = L_{n+1} + L_n$ for $n \ge 1.$ How many terms in the sequence $L_1, L_2, L_3, \ldots, L_{2023}$ are even?

$673$

$1011$

$675$

$1010$

$674$

Answer: E

Concepts:parity recursion pattern recognition

Difficulty rating: 1250

Solution:

Track the parities: $1, 3, 4, 7, 11, 18, \ldots$ run odd, odd, even, then repeat with period $3.$ So $L_n$ is even exactly when $3 \mid n.$ Among $1 \le n \le 2023,$ that's $\lfloor 2023/3 \rfloor = 674$ multiples of $3.$ Therefore, the answer is E.

← Problem 5#5

Full Exam

Problem 7#7 →

Problem 6 in Other Years

2000 AMC 10 · 2001 AMC 10 · 2002 AMC 10A · 2002 AMC 10B · 2003 AMC 10A · 2003 AMC 10B · 2004 AMC 10A · 2004 AMC 10B · 2005 AMC 10A · 2005 AMC 10B · 2006 AMC 10A · 2006 AMC 10B · 2007 AMC 10A · 2007 AMC 10B · 2008 AMC 10A · 2008 AMC 10B · 2009 AMC 10A · 2009 AMC 10B · 2010 AMC 10A · 2010 AMC 10B · 2011 AMC 10A · 2011 AMC 10B · 2012 AMC 10A · 2012 AMC 10B · 2013 AMC 10A · 2013 AMC 10B · 2014 AMC 10A · 2014 AMC 10B · 2015 AMC 10A · 2015 AMC 10B · 2016 AMC 10A · 2016 AMC 10B · 2017 AMC 10A · 2017 AMC 10B · 2018 AMC 10A · 2018 AMC 10B · 2019 AMC 10A · 2019 AMC 10B · 2020 AMC 10A · 2020 AMC 10B · 2021 AMC 10A Spring · 2021 AMC 10B Spring · 2021 AMC 10A Fall · 2021 AMC 10B Fall · 2022 AMC 10A · 2022 AMC 10B · 2023 AMC 10A · 2024 AMC 10A · 2024 AMC 10B · 2025 AMC 10A · 2025 AMC 10B