Difference between revisions of "2022 AMC 12B Problems/Problem 11"
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== Solution 3 (Difference Relation) == | == Solution 3 (Difference Relation) == | ||
Notice that this is a recurrence relationship where <math>a_0=2,a_1=-1,a_n=-a_{n-1}-a_{n-2}</math>. This recurrence relationship goes like <math>2,-1,-1,2,-1,-1,2,\ldots</math>. Every time <math>n</math> is multiple of 3 as is true when <math>n=2022</math>, <math>a_n= \boxed{\textbf{(E)} \ 2}</math> | Notice that this is a recurrence relationship where <math>a_0=2,a_1=-1,a_n=-a_{n-1}-a_{n-2}</math>. This recurrence relationship goes like <math>2,-1,-1,2,-1,-1,2,\ldots</math>. Every time <math>n</math> is multiple of 3 as is true when <math>n=2022</math>, <math>a_n= \boxed{\textbf{(E)} \ 2}</math> | ||
+ | ~lopkiloinm | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2022|ab=B|num-b=10|num-a=12}} | {{AMC12 box|year=2022|ab=B|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 03:55, 19 November 2022
Contents
Problem
Let , where
. What is
?
Solution 1
Converting both summands to exponential form,
Notice that both are scaled copies of the third roots of unity.
When we replace the summands with their exponential form, we get
When we substitute
, we get
We can rewrite
as
, how does that help?
Since any third root of unity must cube to
.
~
Solution 2 (Eisenstein Units)
The numbers and
are both
(along with
), denoted as
and
, respectively. They have the property that when they are cubed, they equal to
. Thus, we can immediately solve:
~mathboy100
Solution 3 (Difference Relation)
Notice that this is a recurrence relationship where . This recurrence relationship goes like
. Every time
is multiple of 3 as is true when
,
~lopkiloinm
See Also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.