If ∑_n=0^∞ cos^2n θ = 5, what is the value of cos 2θ?

2004 AMC 12A Problem 21

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21.

\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}

The series is geometric with ratio

\cos^2 \theta

Sum to get

\dfrac{1}{\sin^2 \theta} = 5,

then use

\cos 2\theta = 1 - 2\sin^2 \theta

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.