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Question 1209803: Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio.
Found 2 solutions by CPhill, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**1. Sum of the Infinite Geometric Series**
* The sum of an infinite geometric series is given by: S = a / (1 - r)
* We are given that the sum is 9: a / (1 - r) = 9
**2. Sum of the Cubes of the Terms**
* The terms of the series are: a, ar, ar², ar³, ...
* Cubing each term: a³, a³r³, a³r⁶, a³r⁹, ...
* This is also a geometric series with:
* First term: a³
* Common ratio: r³
* The sum of this series is: a³ / (1 - r³)
* We are given that this sum is 36: a³ / (1 - r³) = 36
**3. Solve the System of Equations**
* From a / (1 - r) = 9, we get: a = 9(1 - r)
* Substitute this into a³ / (1 - r³) = 36:
* [9(1 - r)]³ / (1 - r³) = 36
* 729(1 - r)³ / (1 - r³) = 36
* (1 - r)³ / (1 - r³) = 36 / 729 = 4 / 81
* Factor (1 - r³):
* (1 - r³) = (1 - r)(1 + r + r²)
* Substitute this into the equation:
* (1 - r)³ / [(1 - r)(1 + r + r²)] = 4 / 81
* (1 - r)² / (1 + r + r²) = 4 / 81
* Take the square root of both sides:
* (1 - r) / √(1 + r + r²) = ± 2 / 9
* We'll assume r is positive, so we'll take the positive root.
* (1 - r) / √(1 + r + r²) = 2 / 9
* 9(1 - r) = 2√(1 + r + r²)
* 81(1 - r)² = 4(1 + r + r²)
* 81(1 - 2r + r²) = 4 + 4r + 4r²
* 81 - 162r + 81r² = 4 + 4r + 4r²
* 77r² - 166r + 77 = 0
* Solve the quadratic equation:
* r = [166 ± √(166² - 4 * 77 * 77)] / (2 * 77)
* r = [166 ± √(27556 - 23716)] / 154
* r = [166 ± √3840] / 154
* r = [166 ± 62.0967] / 154
* We have two possible values for r:
* r1 = (166 + 62.0967) / 154 ≈ 1.481
* r2 = (166 - 62.0967) / 154 ≈ 0.675
* Since the series is convergent, we must have |r| < 1. Therefore, r = 0.675.
**4. Check the Solution**
* r = 0.675 = 27/40
* a = 9(1 - r) = 9(1 - 27/40) = 9(13/40) = 117/40
* a³ / (1 - r³) = (117/40)³ / (1 - (27/40)³) ≈ 36
**Final Answer:** The common ratio is 27/40 or 0.675.
Answer by greenestamps(13200)  (Show Source):
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