Skip to main content

2023 AMC 10B Problem 6

Problem 6 of 25EasierAlgebraNumber Theory

Let L1=1,L_1 = 1, L2=3,L_2 = 3, and Ln+2=Ln+1+LnL_{n+2} = L_{n+1} + L_n for n≥1.n \ge 1. How many terms in the sequence L1,L_1, L2,L_2, L3,L_3, …,\ldots, L2023L_{2023} are even?

Answer choices

Show solution

Solution

Track the parities: 1,3,4,7,11,18,…1, 3, 4, 7, 11, 18, \ldots run odd, odd, even, then repeat with period 3.3. So LnL_n is even exactly when nn is divisible by 3.3. Among 1≤n≤2023,1 \le n \le 2023, that’s ⌊20233⌋=674\lfloor \frac{2023}{3} \rfloor = 674 multiples of 3.3. Therefore, the answer is E.

More practice

Concepts: parity · recursion · pattern recognition

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.