ACT Math Practice Question #851 (Hard (36)) | Test Citadel
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ACT Math Difficulty: Hard (36)

Digital ACT Math Practice Question #851

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

The first term of an infinite geometric series is 18, and the sum of the series is 54. What is the common ratio \( r \) of the series?
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Socratic AI Engine Step-by-Step Derivation
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Tactical Insight (Hint 1)

Set up the infinite sum formula: \( S_\infty = \frac{a_1}{1 - r} \).

Elimination Framework (Hint 2)

Divide: \( 1 - r = 18/54 = 1/3 \). Then subtract from 1 to solve for \( r \).

Masterclass Solution & Distractor Trap Analysis

The sum of an infinite geometric series is given by \( S_\infty = \frac{a_1}{1 - r} \) for \( |r| < 1 \). Given \( a_1 = 18 \) and \( S_\infty = 54 \): \[ 54 = \frac{18}{1 - r} \implies 1 - r = \frac{18}{54} = \frac{1}{3} \] \[ r = 1 - \fra...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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