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Find the sum of the following infinite geometric series, or state that it is not possible. 31 k 8

Find the sum of the following infinite geometric series, or state that it is not possible. 31 k 8

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem Statement:** Find the sum of the following infinite geometric series, or state that it is not possible. \[ \sum_{k=1}^{\infty} 8 \left(\frac{-3}{4}\right)^k \] _To solve this problem:_ 1. **Identify the First Term (a) and Common Ratio (r):** - The first term (a) can be identified as the first term of the series when \(k=1\). - The common ratio (r) is the factor by which each term of the series is multiplied to get the next term. 2. **Geometric Series Sum Formula for |r| < 1:** - For an infinite geometric series to have a sum, the absolute value of the common ratio |r| must be less than 1. - The sum \(S\) of an infinite geometric series is given by: \[ S = \frac{a}{1 - r} \] 3. **Apply the Values:** - Here, the first term \(a_1 = 8 \left(\frac{-3}{4}\right)^1 = 8 \times \frac{-3}{4} = -6\) - The common ratio \(r = \frac{-3}{4}\) 4. **Check if |r| < 1:** - Here, \(|\frac{-3}{4}| = \frac{3}{4} < 1\) 5. **Calculate the Sum:** - Substitute the values into the formula: \[ S = \frac{a}{1 - r} = \frac{-6}{1 - \left(\frac{-3}{4}\right)} = \frac{-6}{1 + \frac{3}{4}} = \frac{-6}{\frac{7}{4}} = -6 \times \frac{4}{7} = \frac{-24}{7} \] Thus, the sum of the given infinite geometric series is: \[ \boxed{\frac{-24}{7}} \]
Transcribed Image Text:**Problem Statement:** Find the sum of the following infinite geometric series, or state that it is not possible. \[ \sum_{k=1}^{\infty} 8 \left(\frac{-3}{4}\right)^k \] _To solve this problem:_ 1. **Identify the First Term (a) and Common Ratio (r):** - The first term (a) can be identified as the first term of the series when \(k=1\). - The common ratio (r) is the factor by which each term of the series is multiplied to get the next term. 2. **Geometric Series Sum Formula for |r| < 1:** - For an infinite geometric series to have a sum, the absolute value of the common ratio |r| must be less than 1. - The sum \(S\) of an infinite geometric series is given by: \[ S = \frac{a}{1 - r} \] 3. **Apply the Values:** - Here, the first term \(a_1 = 8 \left(\frac{-3}{4}\right)^1 = 8 \times \frac{-3}{4} = -6\) - The common ratio \(r = \frac{-3}{4}\) 4. **Check if |r| < 1:** - Here, \(|\frac{-3}{4}| = \frac{3}{4} < 1\) 5. **Calculate the Sum:** - Substitute the values into the formula: \[ S = \frac{a}{1 - r} = \frac{-6}{1 - \left(\frac{-3}{4}\right)} = \frac{-6}{1 + \frac{3}{4}} = \frac{-6}{\frac{7}{4}} = -6 \times \frac{4}{7} = \frac{-24}{7} \] Thus, the sum of the given infinite geometric series is: \[ \boxed{\frac{-24}{7}} \]
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