Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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**Problem Statement:** 1. **Write 0.17̅ as an infinite geometric series using summation notation.** \[ \sum_{{k = 1}}^{\infty} \text{\_\_\_\_\_} \] 2. **Use the formula for finding the sum of an infinite geometric series to convert 0.17̅ to a fraction.** \[ \text{\_\_\_\_\_} \] **Explanation:** - **Infinite Geometric Series:** An infinite geometric series is a series of the form \(a + ar + ar^2 + ar^3 + \ldots\) where \(a\) is the first term and \(r\) is the common ratio. - **Summation Notation:** The series can be expressed in compact notation using the Greek letter sigma (Σ), indicating that you are summing up terms. The expression below the sigma indicates the starting index, and the expression above indicates the ending index (which is infinite in this case). - **Periodic Decimal 0.17̅:** The notation "0.17̅" represents a repeating decimal where "17" continues indefinitely (0.171717...). - **Converting Decimal to Fraction:** To convert repeating decimals to fractions, find the sum of the corresponding infinite geometric series using the formula: \[ S = \frac{a}{1 - r} \] where \(a\) is the first term of the series and \(r\) is the common ratio.
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Transcribed Image Text:**Problem Statement:** 1. **Write 0.17̅ as an infinite geometric series using summation notation.** \[ \sum_{{k = 1}}^{\infty} \text{\_\_\_\_\_} \] 2. **Use the formula for finding the sum of an infinite geometric series to convert 0.17̅ to a fraction.** \[ \text{\_\_\_\_\_} \] **Explanation:** - **Infinite Geometric Series:** An infinite geometric series is a series of the form \(a + ar + ar^2 + ar^3 + \ldots\) where \(a\) is the first term and \(r\) is the common ratio. - **Summation Notation:** The series can be expressed in compact notation using the Greek letter sigma (Σ), indicating that you are summing up terms. The expression below the sigma indicates the starting index, and the expression above indicates the ending index (which is infinite in this case). - **Periodic Decimal 0.17̅:** The notation "0.17̅" represents a repeating decimal where "17" continues indefinitely (0.171717...). - **Converting Decimal to Fraction:** To convert repeating decimals to fractions, find the sum of the corresponding infinite geometric series using the formula: \[ S = \frac{a}{1 - r} \] where \(a\) is the first term of the series and \(r\) is the common ratio.
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