Write the repeating decimal as a fraction. .004004004 ______

Identify Repeating Pattern: Identify the repeating pattern in the decimal. Pattern identified: 0.004004004... = 0.004 + 0.000004 + 0.000000004 + ...Express Terms as Fractions: Express each term in the pattern as a fraction. 0.004004004... = 0.004 + 0.000004 + 0.000000004 + ... = 4/1000 + 4/1000000 + 4/1000000000 + ...Recognize Geometric Series: Recognize that the series 4/1000 + 4/1000000 + 4/1000000000 + ... forms a geometric series. Find the common ratio (r) of the geometric series by dividing two consecutive terms. (4/1000000) / (4/1000) = 4/1000000 * 1000/4 = 1/1000 Common Ratio (r): 1/1000Write as Fraction: Write the repeating decimal as a fraction using the formula for the sum of an infinite geometric series, which is a_1 / (1 - r), where a_1 is the first term. Substitute a_1 = 4/1000 and r = 1/1000 into the formula. (4/1000) / (1 - 1/1000) = (4/1000) / (999/1000) = 4/1000 * 1000/999Simplify Fraction: Simplify the fraction. 4/1000 * 1000/999 = 4/999 So, 0.004004004... = 4/999

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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

.004004004

Identify Repeating Pattern: Identify the repeating pattern in the decimal. $\newline$ Pattern identified: $0.004004004\ldots = 0.004 + 0.000004 + 0.000000004 + \ldots$
Express Terms as Fractions: Express each term in the pattern as a fraction. $\newline$ $0.004004004... = 0.004 + 0.000004 + 0.000000004 + ...$ $\newline$ $= \frac{4}{1000} + \frac{4}{1000000} + \frac{4}{1000000000} + ...$
Recognize Geometric Series: Recognize that the series $\frac{4}{1000} + \frac{4}{1000000} + \frac{4}{1000000000} + \ldots$ forms a geometric series. $\newline$ Find the common ratio $(r)$ of the geometric series by dividing two consecutive terms. $\newline$ $(\frac{4}{1000000}) / (\frac{4}{1000}) = \frac{4}{1000000} \times \frac{1000}{4} = \frac{1}{1000}$ $\newline$ Common Ratio $(r): \frac{1}{1000}$
Write as Fraction: Write the repeating decimal as a fraction using the formula for the sum of an infinite geometric series, which is $a_1 / (1 - r)$ , where $a_1$ is the first term. $\newline$ Substitute $a_1 = \frac{4}{1000}$ and $r = \frac{1}{1000}$ into the formula. $\newline$ $(\frac{4}{1000}) / (1 - \frac{1}{1000}) = (\frac{4}{1000}) / (\frac{999}{1000}) = \frac{4}{1000} \times \frac{1000}{999}$
Simplify Fraction: Simplify the fraction. $\newline$ $\frac{4}{1000} \times \frac{1000}{999} = \frac{4}{999}$ $\newline$ So, $0.004004004\ldots = \frac{4}{999}$