Q. 77 Express each of the repeating de... [FREE SOLUTION]

Chapter 7: Q. 77 (page 616)

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.
$0.199999 . . .$

Short Answer

Expert verified

The given repeating decimal as a geometric series is $\frac{1 + {鈭�}_{k = 0}^{鈭�} 0.9 {(0.1)}^{k}}{10},$ and as the quotient of two integers reduced to lowest terms is $\frac{1}{5} .$

Step by step solution

Step 1. Given Information.

The given repeating decimal is $0.199999 . . .$

Step 2. Express the repeating decimal as a geometric series.

The given repeating decimal starts repeating after the tenths place so, to express it as a geometric series, let $y = 0.199999 . . .$

Now, multiply both the sides by10

$(10) y = (10) 0.199999 . . . 10 y = 1.99999 . . . y = \frac{1.99999 . . .}{10} y = \frac{1 + 0.99999 . . .}{10} y = \frac{1 + 0.9 (1 + 0.1 + {(0.1)}^{2} + {(0.1)}^{3} + . . .)}{10} y = \frac{1 + {鈭�}_{k = 0}^{鈭�} 0.9 {(0.1)}^{k}}{10} . . . . . . . . . . (i)$

Step 3. Express the repeating decimal as the quotient of two integers reduced to the lowest terms.

The given repeating decimal as the quotient of two integers reduced to the lowest terms can be expressed as

$0.199999 . . . = \frac{1 + {鈭�}_{k = 0}^{鈭�} 0.9 {(0.1)}^{k}}{10} Use (i) Now, use S_{鈭�} = \frac{a}{1 - r} = \frac{1 + \frac{0.9}{1 - 0.1}}{10} = \frac{1 + \frac{0.9}{0.9}}{10} = \frac{2}{10} = \frac{1}{5}$

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91影视

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.
$0.199999 . . .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Express the repeating decimal as a geometric series.

Step 3. Express the repeating decimal as the quotient of two integers reduced to the lowest terms.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.0.199999...

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Express the repeating decimal as a geometric series.

Step 3. Express the repeating decimal as the quotient of two integers reduced to the lowest terms.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Logic and Functions

Geometry

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.
$0.199999 . . .$