2004 AMC 12A Problem 21

Problem 21 of 25HarderAlgebra Geometry

\sum_{n=0}^{\infty} \cos^{2n} \theta = 5,

what is the value of

\cos 2\theta?

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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.

Concepts: geometric sequence · trigonometric identity

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