2023 AMC 10B Problem 6

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

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

Even terms occur exactly at positions that are multiples of

3

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.

Problem 6 in other years