Write the repeating decimal as a fraction. .833833833 ______

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating pattern is 833.Express Decimal as Sum: Express the repeating decimal as a sum of its parts: 0.833833833... = 0.833 + 0.000833 + 0.000000833 + ...Convert to Fractions: Convert each part into a fraction: 0.833 = 833/1000, and each subsequent part is 1000 times smaller than the previous one.Recognize Geometric Series: Recognize that this is a geometric series with the first term a_1 = 833/1000 and the common ratio r = 1/1000.Use Infinite Series Formula: Use the formula for the sum of an infinite geometric series, S = a_1 / (1 - r), to find the fraction equivalent of the repeating decimal.Substitute Values: Substitute the values of a_1 and r into the formula: S = (833/1000) / (1 - 1/1000).Simplify Denominator: Simplify the denominator: 1 - 1/1000 = 999/1000.Calculate the Sum: Now, calculate the sum: S = (833/1000) / (999/1000) = 833/1000 * 1000/999.Simplify the Fraction: Simplify the fraction: S = 833/999.

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Write the repeating decimal as a fraction. $\newline$ $.833833833$

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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

.833833833

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating pattern is $833$ .
Express Decimal as Sum: Express the repeating decimal as a sum of its parts: $0.833833833\ldots = 0.833 + 0.000833 + 0.000000833 + \ldots$
Convert to Fractions: Convert each part into a fraction: $0.833 = \frac{833}{1000}$ , and each subsequent part is $1000$ times smaller than the previous one.
Recognize Geometric Series: Recognize that this is a geometric series with the first term $a_1 = \frac{833}{1000}$ and the common ratio $r = \frac{1}{1000}$ .
Use Infinite Series Formula: Use the formula for the sum of an infinite geometric series, $S = \frac{a_1}{1 - r}$ , to find the fraction equivalent of the repeating decimal.
Substitute Values: Substitute the values of $a_1$ and $r$ into the formula: $S = \frac{833}{1000} / \left(1 - \frac{1}{1000}\right)$ .
Simplify Denominator: Simplify the denominator: $1 - \frac{1}{1000} = \frac{999}{1000}$ .
Calculate the Sum: Now, calculate the sum: $S = \frac{833}{1000} / \frac{999}{1000} = \frac{833}{1000} \times \frac{1000}{999}$ .
Simplify the Fraction: Simplify the fraction: $S = \frac{833}{999}$ .