2004 AMC 12A Problem 21 — Solution | LIVE by Po-Shen Loh

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2004 AMC 12A Problem 21

Attempt Problem 21 of the 2004 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12A solutions, or check the answer key.

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21.

If $\sum_{n=0}^{\infty} \cos^{2n} \theta = 5,$ what is the value of $\cos 2\theta?$

$\dfrac{1}{5}$

$\dfrac{2}{5}$

$\dfrac{\sqrt{5}}{5}$

$\dfrac{3}{5}$

$\dfrac{4}{5}$

Answer: D

Concepts:geometric sequence trigonometric identity

Difficulty rating: 1820

Solution:

The series is geometric with first term $1$ and ratio $\cos^2 \theta,$ so its sum is $\dfrac{1}{1 - \cos^2 \theta} = \dfrac{1}{\sin^2 \theta} = 5.$

Thus $\sin^2 \theta = \tfrac15,$ and $\begin{aligned} \cos 2\theta &= 1 - 2\sin^2 \theta \\ &= 1 - \tfrac25 = \tfrac35. \end{aligned}$

Thus, the correct answer is D.

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