2021 AMC 12A Problems/Problem 8
Revision as of 06:17, 29 September 2021 by MRENTHUSIASM (talk | contribs) (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.)
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:
![\[\begin{array}{c||c|c|c|c|c|c|c|c|c|c|c} &&&&&&&&&&& \\ [-2.5ex] \textbf{Term} &\boldsymbol{D_0}&\boldsymbol{D_1}&\boldsymbol{D_2}&\boldsymbol{D_3}&\boldsymbol{D_4}&\boldsymbol{D_5}&\boldsymbol{D_6}&\boldsymbol{D_7}&\boldsymbol{D_8}&\boldsymbol{D_9}&\boldsymbol{\cdots} \\ \hline \hline &&&&&&&&&&& \\ [-2.25ex] \textbf{Value} & 0&0&1&1&1&2&3&4&6&9&\cdots \\ \hline &&&&&&&&&&& \\ [-2.25ex] \textbf{Parity} & E&E&O&O&O&E&O&E&E&O&\cdots \end{array}\]](http://latex.artofproblemsolving.com/c/6/6/c66194484f13cb67490c65bf30cd239601814333.png)
Note that 
have the same parities as 
so the parity is periodic with period 
Since the remainders of 
are 
the parities of are the same as the parities of 
namely 
~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)
~ 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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