Write the repeating decimal as a fraction. .557557557 ______

Identify Repeating Pattern: Let's first identify the repeating pattern in the decimal. The digits 557 are repeating. Pattern identified: 0.557557557... = 0.557 + 0.000557 + 0.000000557 + ...Express as Sum of Fractions: Now, let's express the repeating decimal as a sum of fractions. 0.557557557... = 557/1000 + 557/1000000 + 557/1000000000 + ...Use Geometric Series Formula: We can see that this is a geometric series with the first term a_1 = 557/1000 and the common ratio r = 1/1000. To find the sum of an infinite geometric series, we use the formula S = a_1 / (1 - r).Substitute Values into Formula: Substitute the values of a_1 and r into the formula. S = (557/1000) / (1 - 1/1000) S = (557/1000) / (999/1000)Simplify Fraction: Now, simplify the fraction by multiplying the numerator and the denominator by the reciprocal of the denominator. S = (557/1000) * (1000/999) S = 557/999

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Write the repeating decimal as a fraction. $\newline$ $.557557557$

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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

.557557557

Identify Repeating Pattern: Let's first identify the repeating pattern in the decimal. The digits $557$ are repeating. $\newline$ Pattern identified: $0.557557557\ldots = 0.557 + 0.000557 + 0.000000557 + \ldots$
Express as Sum of Fractions: Now, let's express the repeating decimal as a sum of fractions. $0.557557557\ldots = \frac{557}{1000} + \frac{557}{1000000} + \frac{557}{1000000000} + \ldots$
Use Geometric Series Formula: We can see that this is a geometric series with the first term $a_1 = \frac{557}{1000}$ and the common ratio $r = \frac{1}{1000}$ . To find the sum of an infinite geometric series, we use the formula $S = \frac{a_1}{(1 - r)}$ .
Substitute Values into Formula: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ $S = \frac{557}{1000} / \left(1 - \frac{1}{1000}\right)$ $\newline$ $S = \frac{557}{1000} / \frac{999}{1000}$
Simplify Fraction: Now, simplify the fraction by multiplying the numerator and the denominator by the reciprocal of the denominator. $\newline$ $S = \frac{557}{1000} \times \frac{1000}{999}$ $\newline$ $S = \frac{557}{999}$