Question

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

Small Hint:

Track only the parity of the terms: odd, odd, even, and then it repeats

Big Hint:

Even terms occur exactly at positions that are multiples of $3$

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 $n$ is divisible by $3.$ Among $1 \le n \le 2023,$ that’s $\lfloor \frac{2023}{3} \rfloor = 674$ multiples of $3.$ Therefore, the answer is E.

2023 AMC 10B Problem 6

Answer: E

Difficulty rating: 1250

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years