Write the repeating decimal as a fraction. .002002002 ______

Identify Repeating Pattern: Identify the repeating pattern in the decimal. The repeating pattern is 002, which repeats indefinitely after the decimal point.Express as Infinite Sum: Express the repeating decimal as an infinite sum of its repeating parts. 0.002002002... = 0.002 + 0.000002 + 0.000000002 + ...Convert to Fractions: Convert each term of the sum into a fraction. 0.002 = 2/1000, 0.000002 = 2/1000000, 0.000000002 = 2/1000000000, and so on.Recognize Geometric Series: Recognize that the sum forms a geometric series with the first term a_1 = 2/1000 and the common ratio r = 1/1000.Use Sum Formula: Use the formula for the sum of an infinite geometric series, S = a_1 / (1 - r), to write the repeating decimal as a fraction. Substitute a_1 = 2/1000 and r = 1/1000 into the formula. S = (2/1000) / (1 - 1/1000)Simplify Expression: Simplify the expression by performing the operations. S = (2/1000) / (999/1000) S = 2/1000 * 1000/999 S = 2/999

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Write the repeating decimal as a fraction. $\newline$ $.002002002$

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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.

\newline

.002002002

Identify Repeating Pattern: Identify the repeating pattern in the decimal. $\newline$ The repeating pattern is $002$ , which repeats indefinitely after the decimal point.
Express as Infinite Sum: Express the repeating decimal as an infinite sum of its repeating parts. $\newline$ $0.002002002\ldots = 0.002 + 0.000002 + 0.000000002 + \ldots$
Convert to Fractions: Convert each term of the sum into a fraction. $\newline$ $0.002 = \frac{2}{1000}$ , $0.000002 = \frac{2}{1000000}$ , $0.000000002 = \frac{2}{1000000000}$ , and so on.
Recognize Geometric Series: Recognize that the sum forms a geometric series with the first term $a_1 = \frac{2}{1000}$ and the common ratio $r = \frac{1}{1000}$ .
Use Sum Formula: Use the formula for the sum of an infinite geometric series, $S = \frac{a_1}{1 - r}$ , to write the repeating decimal as a fraction. $\newline$ Substitute $a_1 = \frac{2}{1000}$ and $r = \frac{1}{1000}$ into the formula. $\newline$ $S = \frac{\left(\frac{2}{1000}\right)}{1 - \frac{1}{1000}}$
Simplify Expression: Simplify the expression by performing the operations. $\newline$ $S = \frac{2}{1000} / \frac{999}{1000}$ $\newline$ $S = \frac{2}{1000} \times \frac{1000}{999}$ $\newline$ $S = \frac{2}{999}$