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Difference between revisions of "2021 AMC 12A Problems/Problem 8"

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==Problem==
 
==Problem==
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A sequence of numbers is defined by <math>D_0=0,D_0=1,D_2=1</math> and <math>D_n=D_{n-1}+D_{n-3}</math> for <math>n\ge 3</math>. What are the parities (evenness or oddness) of the triple of numbers <math>(D_{2021},D_{2022},D_{2023})</math>, where <math>E</math> denotes even and <math>O</math> denotes odd?
+
A sequence of numbers is defined by <math>D_0=0,D_1=0,D_2=1</math> and <math>D_n=D_{n-1}+D_{n-3}</math> for <math>n\ge 3</math>. What are the parities (evenness or oddness) of the triple of numbers <math>(D_{2021},D_{2022},D_{2023})</math>, where <math>E</math> denotes even and <math>O</math> denotes odd?
  
 
<math>\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)</math>
 
<math>\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)</math>
  
 
==Solution==
 
==Solution==
−
Making a small chart, we have
+
We construct the following table:
−
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
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<cmath>\begin{array}{c||c|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
+
&&&&&&&&&&& \\ [-2.5ex]
−
0&0&1&1&1&2&3&4&6&9\\\hline
+
\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} \\
−
E&E&O&O&O&E&O&E&E&O
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\hline \hline
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\end{tabular}
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&&&&&&&&&&& \\ [-2.25ex]
−
This starts repeating every 7 terms, so <math>D_{2021}=D_5=</math> E, <math>D_{2022}=D_6=</math> O, and <math>D_{2023}=D_7=</math> E. Thus, the answer is \boxed{C) \text{(E, O, E)}}
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\textbf{Value} & 0&0&1&1&1&2&3&4&6&9&\cdots \\ \hline
−
~JHawk0224
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&&&&&&&&&&& \\ [-2.25ex]
 +
\textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots
 +
\end{array}</cmath>
 +
Note that <math>(D_7,D_8,D_9)</math> and <math>(D_0,D_1,D_2)</math> have the same parities, so the parity is periodic with period <math>7.</math> Since the remainders of <math>(2021\div7,2022\div7,2023\div7)</math> are <math>(5,6,7),</math> we conclude that <math>(D_{2021},D_{2022},D_{2023})</math> and <math>(D_5,D_6,D_7)</math> have the same parities, namely <math>\boxed{\textbf{(C) }(E,O,E)}.</math>
 +
 
 +
~JHawk0224 ~MRENTHUSIASM
 +
 
 +
==Video Solution (Quick and Easy)==
 +
https://youtu.be/ecLkESGj-pY
 +
 
 +
~Education, the Study of Everything
 +
 
 +
 
 +
==Video Solution by Aaron He (Finding Cycles)==
 +
https://www.youtube.com/watch?v=xTGDKBthWsw&t=7m43s
 +
 
 +
==Video Solution by Hawk Math==
 +
https://www.youtube.com/watch?v=P5al76DxyHY
 +
 
 +
== Video Solution by OmegaLearn (Using Parity and Pattern Finding) ==
 +
https://youtu.be/TSBjbhN_QKY
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/cckGBU2x1zg?t=227
 +
 
 +
~IceMatrix
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2021|ab=A|num-b=7|num-a=9}}
 
{{AMC12 box|year=2021|ab=A|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 22:25, 22 October 2022

Problem

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?

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

Solution

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 $\boxed{\textbf{(C) }(E,O,E)}.$

~JHawk0224 ~MRENTHUSIASM

Video Solution (Quick and Easy)

https://youtu.be/ecLkESGj-pY

~Education, the Study of Everything


Video Solution by Aaron He (Finding Cycles)

https://www.youtube.com/watch?v=xTGDKBthWsw&t=7m43s

Video Solution by Hawk Math

https://www.youtube.com/watch?v=P5al76DxyHY

Video Solution by OmegaLearn (Using Parity and Pattern Finding)

https://youtu.be/TSBjbhN_QKY

~ pi_is_3.14

Video Solution by TheBeautyofMath

https://youtu.be/cckGBU2x1zg?t=227

~IceMatrix

See also

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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