AMC 10 · 2004 · #21

Grade 11 algebra
Archetype Geometric Series Share └ Sub-archetype Cyclic Participant Share
geometric-series-infinitedouble-angle-formulapythagorean-identity convert-to-algebrapattern-recognition ↑ Prerequisites: sequences-geometric
📏 Medium solution 💡 2 insights
Problem
Add up 1 + cos²θ + cos⁴θ + cos⁶θ and so on forever. That endless sum equals 5. From that fact alone, find cos 2θ.

Pick an answer.

(A)
$\frac15$
(B)
$\frac25$
(C)
$\frac {\sqrt5}{5}$
(D)
$\frac35$
(E)
$\frac45$
How to solve
Strategy Introduce a Variable

The sum looks like a trig problem, but nothing trigonometric is happening inside it: the terms are just (1, u, u², u³, …) for the one number (u = cos²θ). Naming that number converts the whole left side into a plain geometric series and the equation into a one-line algebra problem for (u). Trigonometry only comes back at the very end, to turn (cos²θ) into (cos 2θ) with the double-angle identity.

1STEP 1

See the powers of one number

Each term is a power of one number, so this is a geometric series.

cos²ⁿθ = (cos²θ)ⁿ = uⁿ, Σ_n=0^∞cos²ⁿθ = 1 + u + u² + u³ + …
2STEP 2

Sum it, and check it converges

The finite sum forces the ratio below one, so the closed form is legal.

Σ_n=0^N uⁿ = (1-u^N+1)/(1-u) ⟶ 1/(1-u) (N → ∞, 0 ≤ u ＜ 1)
3STEP 3

Solve for cos squared theta

Solving the resulting linear equation gives cos²θ = 4/5.

1/(1-u) = 5 ⟹ 1 - u = 1/5 ⟹ u = cos²θ = 4/5
4STEP 4

Expand the double angle

The double angle expands to cos²θ - sin²θ.

cos 2θ = cos(θ+θ) = cosθcosθ - sinθsinθ = cos²θ - sin²θ
5STEP 5

Substitute and finish

Rewriting with the identity and substituting gives 3/5, choice (D).

cos 2θ = cos²θ - (1 - cos²θ) = 2cos²θ - 1 = 2 · 4/5 - 1 = 3/5
Answer
3/5
Check the answer is actually achievable. (cos²θ = 4/5) lies in ([0,1)), so a real angle exists: (θ = arccos2/√5 ≈ 26.57°). Then (2θ ≈ 53.13°) and (cos 2θ ≈ 0.6 = 3/5). Summing numerically also works: (1 + 0.8 + 0.64 + 0.512 + … = 1/0.2 = 5). Two of the wrong choices are the leftovers of the computation and are worth naming: (4/5) is (cos²θ) itself (stopping one step early) and (1/5) is (sin²θ). Both are values that appear on the way, neither is what was asked.
💡Key takeaway

When every term of an endless sum is the same number multiplied on again, the whole thing collapses to (1/(1 - that number)).

  • See the powers of one number
  • Sum it, and check it converges
  • Solve for cos squared theta
  • Expand the double angle
  • Substitute and finish

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