Difference between revisions of "2022 AMC 12B Problems/Problem 11"
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== Solution 3 (Linear Second-order Homogeneous Difference Equation) == | == Solution 3 (Linear Second-order Homogeneous Difference Equation) == | ||
− | Notice how this looks like the closed form of the Fibonacci sequence. The base of the exponents are the roots of the characteristic equation <math>r^2+r+1=0</math>. So we have <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 <math>3</math> as is true when <math>n=2022</math>, <math>a_n= \boxed{\textbf{(E)} \ 2}</math> | + | Notice how this looks like the closed form of the Fibonacci sequence except different roots. This is motivation to turn this closed formula into a recurrence relation. The base of the exponents are the roots of the characteristic equation <math>r^2+r+1=0</math>. So we have <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 <math>3</math> as is true when <math>n=2022</math>, <math>a_n= \boxed{\textbf{(E)} \ 2}</math> |
~lopkiloinm | ~lopkiloinm | ||
Revision as of 04:07, 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 (Linear Second-order Homogeneous Difference Equation)
Notice how this looks like the closed form of the Fibonacci sequence except different roots. This is motivation to turn this closed formula into a recurrence relation. The base of the exponents are the roots of the characteristic equation . So we have
. This recurrence relationship goes like
. Every time
is multiple of
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.