AMC10 2023 B

AMC10 2023 B · Q6

AMC10 2023 B · Q6. It mainly tests Sequences & recursion (algebra), Parity (odd/even).

Let $L_{1}=1, L_{2}=3$, and $L_{n+2}=L_{n+1}+L_{n}$ for $n\geq 1$. How many terms in the sequence $L_{1}, L_{2}, L_{3},...,L_{2023}$ are even?

设 $L_{1}=1, L_{2}=3$，且 $L_{n+2}=L_{n+1}+L_{n}$ 对于 $n\geq 1$。序列 $L_{1}, L_{2}, L_{3},...,L_{2023}$ 中有多少项是偶数？

(A) 673 673

(B) 1011 1011

(D) 1010 1010

(E) 674 674

Answer

Correct choice: (E)

正确答案：（E）

Solution

We calculate more terms: \[1,3,4,7,11,18,\ldots.\] We find a pattern: if $n+2$ is a multiple of $3$, then the term is even, or else it is odd. There are $\left\lfloor \frac{2023}{3} \right\rfloor =\boxed{\textbf{(E) }674}$ multiples of $3$ from $1$ to $2023$.

我们计算更多项：\[1,3,4,7,11,18,\ldots.\] 我们发现一个模式：如果 $n+2$ 是 3 的倍数，则该项是偶数，否则是奇数。从 1 到 2023 有 $\left\lfloor \frac{2023}{3} \right\rfloor =\boxed{\textbf{(E) }674}$ 个 3 的倍数。

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