2021 AMC 12A Problems/Problem 8

Problem

A sequence of numbers is defined by $D_0=0,D_0=1,D_2=1$ 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?

$\textbf{(A) }(O,E,O) \qquad \textbf{(B) }(E,E,O) \qquad \textbf{(C) }(E,O,E) \qquad \textbf{(D) }(O,O,E) \qquad \textbf{(E) }(O,O,O)$

Solution

Making a small chart, we have \begin{tabular}{c|c|c|c|c|c|c|c|c|c} D_0&D_1&D_2&D_3&D_4&D_5&D_6&D_7&D_8&D_9\\\hline 0&0&1&1&1&2&3&4&6&9\\\hline E&E&O&O&O&E&O&E&E&O \end{tabular} This starts repeating every 7 terms, so $D_{2021}=D_5=$ E, $D_{2022}=D_6=$ O, and $D_{2023}=D_7=$ E. Thus, the answer is \boxed{C) \text{(E, O, E)}} ~JHawk0224

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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All AMC 12 Problems and Solutions

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2021 AMC 12A Problems/Problem 8

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