AMC12 2021 A

AMC12 2021 A · Q8

AMC12 2021 A · Q8. It mainly tests Parity (odd/even), Sequences in number theory (remainders patterns).

A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd?

一个数列由 $D_0=0,D_1=0,D_2=1$ 和 $D_n=D_{n-1}+D_{n-3}$（$n\ge 3$）定义。求三元组 $(D_{2021},D_{2022},D_{2023})$ 的奇偶性，其中 $E$ 表示偶数，$O$ 表示奇数。

(A) (O,E,O) (O,E,O)

(B) (E,E,O) (E,E,O)

(D) (O,O,E) (O,O,E)

(E) (O,O,O) (O,O,O)

Answer

Correct choice: (C)

正确答案：（C）

Solution

We construct the following table: \[\begin{array}{c||c|c|c|c|c|c|c|c|c|c|c} &&&&&&&&&&& \\ [-2.5ex] \textbf{Term} &\boldsymbol{D_0}&\boldsymbol{D_1}&\boldsymbol{D_2}&\boldsymbol{D_3}&\boldsymbol{D_4}&\boldsymbol{D_5}&\boldsymbol{D_6}&\boldsymbol{D_7}&\boldsymbol{D_8}&\boldsymbol{D_9}&\boldsymbol{\cdots} \\ \hline \hline &&&&&&&&&&& \\ [-2.25ex] \textbf{Value} & 0&0&1&1&1&2&3&4&6&9&\cdots \\ \hline &&&&&&&&&&& \\ [-2.25ex] \textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots \end{array}\] Note that $(D_7,D_8,D_9)$ and $(D_0,D_1,D_2)$ have the same parities, so the parity is periodic with period $7.$ Since the remainders of $(2021\div7,2022\div7,2023\div7)$ are $(5,6,7),$ we conclude that $(D_{2021},D_{2022},D_{2023})$ and $(D_5,D_6,D_7)$ have the same parities, namely $\boxed{\textbf{(C) }(E,O,E)}.$

我们构造如下表格： \[\begin{array}{c||c|c|c|c|c|c|c|c|c|c|c} &&&&&&&&&&& \\ [-2.5ex] \textbf{Term} &\boldsymbol{D_0}&\boldsymbol{D_1}&\boldsymbol{D_2}&\boldsymbol{D_3}&\boldsymbol{D_4}&\boldsymbol{D_5}&\boldsymbol{D_6}&\boldsymbol{D_7}&\boldsymbol{D_8}&\boldsymbol{D_9}&\boldsymbol{\cdots} \\ \hline \hline &&&&&&&&&&& \\ [-2.25ex] \textbf{Value} & 0&0&1&1&1&2&3&4&6&9&\cdots \\ \hline &&&&&&&&&&& \\ [-2.25ex] \textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots \end{array}\] 注意 $(D_7,D_8,D_9)$ 与 $(D_0,D_1,D_2)$ 具有相同的奇偶性，因此奇偶性以周期 $7$ 重复。由于 $(2021\div7,2022\div7,2023\div7)$ 的余数为 $(5,6,7)$，我们得出 $(D_{2021},D_{2022},D_{2023})$ 与 $(D_5,D_6,D_7)$ 具有相同的奇偶性，即 $\boxed{\textbf{(C) }(E,O,E)}。$

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