Let f(n) = ((-1 + i\sqrt3)/(2))^n + ((-1 - i\sqrt3)/(2))^n, where i = √-1. What is f(2022)?

2022 AMC 12B Problem 11 — Solution

2022 AMC 12B Problem 11

Below is the professionally curated solution for Problem 11 of the 2022 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12B solutions, or check the answer key.

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Concepts:roots of unity complex number

Difficulty rating: 1570

11.

Let $f(n) = \left(\dfrac{-1 + i\sqrt3}{2}\right)^n + \left(\dfrac{-1 - i\sqrt3}{2}\right)^n,$ where $i = \sqrt{-1}.$ What is $f(2022)?$

$-2$

$-1$

$0$

$\sqrt3$

$2$

Solution:

The two bases are the primitive cube roots of unity, $\omega = e^{2\pi i/3}$ and its conjugate $\omega^2 = e^{-2\pi i/3}.$ So $f(n) = \omega^n + \omega^{-n} = 2\cos\dfrac{2\pi n}{3}.$

Since $2022$ is a multiple of $3,$ $\omega^{2022} = 1,$ so $f(2022) = 1 + 1 = 2.$

Thus, the correct answer is E.

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