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

(1. Fixed the code of the table, so the subscripts are really showing. 2. Minor revisions. Recall that we have to be cautious in the use of equal signs.)
(Solution)
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\textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots
 
\textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots
 
\end{array}</cmath>
 
\end{array}</cmath>
Note that <math>(D_7,D_8,D_9)</math> have the same parities as <math>(D_0,D_1,D_2),</math> 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> the parities of <math>(D_{2021},D_{2022},D_{2023})</math> are the same as the parities of <math>(D_5,D_6,D_7),</math> namely <math>\boxed{\textbf{(C) }(E,O,E)}.</math>
+
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
 
~JHawk0224 ~MRENTHUSIASM

Revision as of 07:46, 29 September 2021

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 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 (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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