Difference between revisions of "2021 AMC 12A Problems/Problem 8"
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==Problem== | ==Problem== | ||
− | A sequence of numbers is defined by <math>D_0=0, | + | 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== | ||
− | + | We construct the following table: | |
+ | <cmath>\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}</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 | |
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− | + | ==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== | ==Video Solution by Hawk Math== | ||
https://www.youtube.com/watch?v=P5al76DxyHY | 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 23:25, 22 October 2022
Contents
Problem
A sequence of numbers is defined by and
for
. What are the parities (evenness or oddness) of the triple of numbers
, where
denotes even and
denotes odd?
Solution
We construct the following table:
Note that
and
have the same parities, so the parity is periodic with period
Since the remainders of
are
we conclude that
and
have the same parities, namely
~JHawk0224 ~MRENTHUSIASM
Video Solution (Quick and Easy)
~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)
~ 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 |
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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