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Difference between revisions of "2022 AMC 12B Problems/Problem 11"

m (→‎Solution 2 (Eisenstein Units))
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\textbf{(E)}\ 2</math>
 
\textbf{(E)}\ 2</math>
  
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== Solution 1 ==
+
== Solution 1 (Quick and Easy) ==
 +
 
 +
We begin by recognizing this form looks similar to the definition of cosine: <cmath>\cos(x)=\frac{e^{ix}+e^{-ix}}{2}.</cmath> We can convert our two terms into exponential form to find <cmath>f(n) = \left( e^{\frac{2\pi i}{3}} \right ) ^n + \left ( e^{-\frac{2\pi i}{3}} \right ) ^n.</cmath> This simplifies nicely: <cmath>e^{\frac{2 \pi i n}{3}} + e^{-\frac{2\pi i n}{3}}=2\cos\left( \frac{2\pi n}{3} \right).</cmath> Thus, <cmath>f(2022)=2\cos \left ( \frac{2\pi (2022) }{3} \right) = 2\cos(1348 \pi) = \boxed{\textbf{(E)}\ 2}.</cmath>
 +
 
 +
~Indiiiigo
 +
 
 +
== Solution 2 ==
 
Converting both summands to exponential form, <cmath>-1 + i\sqrt{3} = 2e^{\frac{2\pi i}{3}}</cmath>
 
Converting both summands to exponential form, <cmath>-1 + i\sqrt{3} = 2e^{\frac{2\pi i}{3}}</cmath>
 
<cmath>-1 - i\sqrt{3} = 2e^{-\frac{2\pi i}{3}} = 2e^{\frac{4\pi i}{3}}</cmath>
 
<cmath>-1 - i\sqrt{3} = 2e^{-\frac{2\pi i}{3}} = 2e^{\frac{4\pi i}{3}}</cmath>
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~ <math>\color{magenta} zoomanTV</math>
 
~ <math>\color{magenta} zoomanTV</math>
  
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== Solution 2 (Eisenstein Units) ==
+
== Solution 3 (Eisenstein Units) ==
 
The numbers <math>\frac{-1+i\sqrt{3}}{2}</math> and <math>\frac{-1-i\sqrt{3}}{2}</math> are both <math>\textbf{Eisenstein Units}</math> (along with <math>1</math>), denoted as <math>\omega</math> and <math>\omega^2</math>, respectively. They have the property that when they are cubed, they equal to <math>1</math>. Thus, we can immediately solve:
 
The numbers <math>\frac{-1+i\sqrt{3}}{2}</math> and <math>\frac{-1-i\sqrt{3}}{2}</math> are both <math>\textbf{Eisenstein Units}</math> (along with <math>1</math>), denoted as <math>\omega</math> and <math>\omega^2</math>, respectively. They have the property that when they are cubed, they equal to <math>1</math>. Thus, we can immediately solve:
  
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~mathboy100
 
~mathboy100
  
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== Solution 3 (Third-order Homogeneous Linear Recurrence Relation) ==
+
== Solution 4 (Third-order Homogeneous Linear Recurrence Relation) ==
 
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^3-1=0</math>. So we have  
 
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^3-1=0</math>. So we have  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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~lopkiloinm
 
~lopkiloinm
  
−
== Solution 4 (Polynomial + Recursion) ==
+
== Solution 5 (Polynomial + Recursion) ==
 
let <math>a = \frac{-1+i\sqrt{3}}{2}</math> and <math>b = \frac{-1-i\sqrt{3}}{2}</math>
 
let <math>a = \frac{-1+i\sqrt{3}}{2}</math> and <math>b = \frac{-1-i\sqrt{3}}{2}</math>
 
<math>a + b = -1</math>
 
<math>a + b = -1</math>

Revision as of 22:38, 20 November 2022

Problem

Let $f(n) = \left( \frac{-1+i\sqrt{3}}{2} \right)^n + \left( \frac{-1-i\sqrt{3}}{2} \right)^n$, where $i = \sqrt{-1}$. What is $f(2022)$?

$\textbf{(A)}\ -2 \qquad \textbf{(B)}\ -1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \sqrt{3} \qquad \textbf{(E)}\ 2$

Solution 1 (Quick and Easy)

We begin by recognizing this form looks similar to the definition of cosine: \[\cos(x)=\frac{e^{ix}+e^{-ix}}{2}.\] We can convert our two terms into exponential form to find \[f(n) = \left( e^{\frac{2\pi i}{3}} \right ) ^n + \left ( e^{-\frac{2\pi i}{3}} \right ) ^n.\] This simplifies nicely: \[e^{\frac{2 \pi i n}{3}} + e^{-\frac{2\pi i n}{3}}=2\cos\left( \frac{2\pi n}{3} \right).\] Thus, \[f(2022)=2\cos \left ( \frac{2\pi (2022) }{3} \right) = 2\cos(1348 \pi) = \boxed{\textbf{(E)}\ 2}.\]

~Indiiiigo

Solution 2

Converting both summands to exponential form, \[-1 + i\sqrt{3} = 2e^{\frac{2\pi i}{3}}\] \[-1 - i\sqrt{3} = 2e^{-\frac{2\pi i}{3}} = 2e^{\frac{4\pi i}{3}}\]

Notice that both are scaled copies of the third roots of unity. When we replace the summands with their exponential form, we get \[f(n) = \left(e^{\frac{2\pi i}{3}}\right)^n + \left(e^{\frac{4\pi i}{3}}\right)^n\] When we substitute $n = 2022$, we get \[f(2022) = \left(e^{\frac{2\pi i}{3}}\right)^{2022} + \left(e^{\frac{4\pi i}{3}}\right)^{2022}\] We can rewrite $2022$ as $3 \cdot 674$, how does that help? \[f(2022) = \left(e^{\frac{2\pi i}{3}}\right)^{3 \cdot 674} + \left(e^{\frac{4\pi i}{3}}\right)^{3 \cdot 674} =\] \[\left(\left(e^{\frac{2\pi i}{3}}\right)^{3}\right)^{674} + \left(\left(e^{\frac{4\pi i}{3}}\right)^{3}\right)^{674} =\] \[1^{674} + 1^{674} = \boxed{\textbf{(E)} \ 2}\] Since any third root of unity must cube to $1$.

~ $\color{magenta} zoomanTV$

Solution 3 (Eisenstein Units)

The numbers $\frac{-1+i\sqrt{3}}{2}$ and $\frac{-1-i\sqrt{3}}{2}$ are both $\textbf{Eisenstein Units}$ (along with $1$), denoted as $\omega$ and $\omega^2$, respectively. They have the property that when they are cubed, they equal to $1$. Thus, we can immediately solve:

\[\omega^{2022} + \omega^{2 \cdot 2022}\] \[= \omega^{3 \cdot 674} + \omega^{3 \cdot 2 \cdot 674}\] \[= 1^{674} + 1^{2 \cdot 674}\] \[= \boxed{\textbf{(E)} \ 2}\]

~mathboy100

Solution 4 (Third-order Homogeneous Linear Recurrence Relation)

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 $r^3-1=0$. So we have \begin{align*} a_0&=2\\ a_1&=-1\\ a_2&=-1\\ a_n&=a_{n-3} \end{align*} Every time $n$ is multiple of $3$ as is true when $n=2022$, $a_n= \boxed{\textbf{(E)} \ 2}$ ~lopkiloinm

Solution 5 (Polynomial + Recursion)

let $a = \frac{-1+i\sqrt{3}}{2}$ and $b = \frac{-1-i\sqrt{3}}{2}$ $a + b = -1$ $a * b = 1$ Therefore a and b are the roots of $x^2 + x + 1 = 0$ By factor theorem $a^2 + a + 1 = 0$ and $b^2 + b + 1 = 0$ Multiply the first equation by $a^{n-2}$ and the second equation by $b^{n-2}$ This gives us $a^n + a^{n-1} + a^{n-2} = 0$ and $b^n + b^{n-1} + b^{n-2} = 0$. Adding both equations together we get $a^n + b^n + a^{n-1} + b^{n-1}+ a^{n-2} + b^{n-2} = 0$ This is the same as $f(n) + f(n-1) + f(n-2) = 0$. Therefore $f(n) = -f(n-1) - f(n-2)$ Plugging in $n=1,2,3,4,5,6$ we get $f(n) = -1, -1, 2, -1, -1, 2$ therefore we know that if $n$ is a multiple of $3$, then $f(n)$ is $2$. Since $2022$ is a multiple of $3$, our answers is $E) 2.$ ~vpeddi18

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

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