Write the repeating decimal as a fraction. .70707070 ______

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating pattern is ＂70＂, which means the decimal can be expressed as 0.70 + 0.0070 + 0.000070 + ... and so on.Express in Fraction Form: Now, let's express each term in fraction form. The decimal 0.70707070... can be written as 70/100 + 70/10000 + 70/1000000 + ... .Recognize Geometric Series: Recognize that this series is a geometric series with the first term a_1 = 70/100 and a common ratio r = 1/100.Use Sum Formula: To find the sum of an infinite geometric series, we use the formula S = a_1 / (1 - r), where S is the sum, a_1 is the first term, and r is the common ratio. Here, a_1 = 70/100 and r = 1/100.Substitute Values: Substitute the values into the formula to get the sum S = (70/100) / (1 - 1/100).Simplify Expression: Simplify the expression: S = (70/100) / (99/100) = 70/99.Final Fraction: Therefore, the repeating decimal 0.70707070... as a fraction is 70/99.

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Algebra 2

Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

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.70707070

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating pattern is ＂ $70$ ＂, which means the decimal can be expressed as $0.70 + 0.0070 + 0.000070 + \ldots$ and so on.
Express in Fraction Form: Now, let's express each term in fraction form. The decimal $0.70707070\ldots$ can be written as $\frac{70}{100} + \frac{70}{10000} + \frac{70}{1000000} + \ldots$ .
Recognize Geometric Series: Recognize that this series is a geometric series with the first term $a_1 = \frac{70}{100}$ and a common ratio $r = \frac{1}{100}$ .
Use Sum Formula: To find the sum of an infinite geometric series, we use the formula $S = \frac{a_1}{1 - r}$ , where $S$ is the sum, $a_1$ is the first term, and $r$ is the common ratio. Here, $a_1 = \frac{70}{100}$ and $r = \frac{1}{100}$ .
Substitute Values: Substitute the values into the formula to get the sum $S = \frac{70}{100} / \left(1 - \frac{1}{100}\right)$ .
Simplify Expression: Simplify the expression: $S = \frac{70}{100} / \frac{99}{100} = \frac{70}{99}$ .
Final Fraction: Therefore, the repeating decimal $0.70707070\ldots$ as a fraction is $\frac{70}{99}$ .