Let f(n) &= ((-1 + i√3)/(2))^n & + ((-1 - i√3)/(2))^n, where i = √-1. What is f(2022)?

2022 AMC 12B Problem 11

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11.

Let

\begin{aligned} f(n) &= \left(\dfrac{-1 + i\sqrt3}{2}\right)^n \\ &\quad {}+ \left(\dfrac{-1 - i\sqrt3}{2}\right)^n, \end{aligned}

where

i = \sqrt{-1}.

What is

f(2022)?

-2

-1

0

\sqrt3

2

The two bases are the primitive cube roots of unity,

\omega

and

\omega^2

Then

f(n) = \omega^n + \omega^{-n};

reduce the exponent

2022

modulo

3

The two bases are the primitive cube roots of unity,

\omega = e^{\frac{2\pi i}{3}}

and its conjugate

\omega^2 = e^{-\frac{2\pi i}{3}}.

f(n) = \omega^n + \omega^{-n} = 2\cos\dfrac{2\pi n}{3}.

Since

2022

is a multiple of

3,

\omega^{2022} = 1,

f(2022) = 1 + 1 = 2.

Thus, the correct answer is E.