Write the repeating decimal as a fraction. .55555555 ______

Identify Repeating Pattern: Identify the repeating pattern in the decimal. The repeating decimal is 0.55555555..., where the digit 5 repeats indefinitely.Express as Infinite Sum: Express the repeating decimal as an infinite sum of its terms. 0.55555555... can be written as 0.5 + 0.05 + 0.005 + ...Convert to Fraction Form: Convert each term into fraction form. 0.55555555... = 5/10 + 5/100 + 5/1000 + ...Recognize Geometric Series: Recognize that the series forms a geometric series with the first term a_1 = 5/10 and a common ratio r = 1/10.Use Infinite Series 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 = 5/10 and r = 1/10 into the formula. S = (5/10) / (1 - 1/10)Simplify Expression: Simplify the expression. S = (5/10) / (9/10) S = 5/10 * 10/9 S = 5/9

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Write the repeating decimal as a fraction. $\newline$ $.55555555$

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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

.55555555

Identify Repeating Pattern: Identify the repeating pattern in the decimal. $\newline$ The repeating decimal is $0.55555555\ldots$ , where the digit $5$ repeats indefinitely.
Express as Infinite Sum: Express the repeating decimal as an infinite sum of its terms. $\newline$ $0.55555555\ldots$ can be written as $0.5 + 0.05 + 0.005 + \ldots$
Convert to Fraction Form: Convert each term into fraction form. $0.55555555\ldots = \frac{5}{10} + \frac{5}{100} + \frac{5}{1000} + \ldots$
Recognize Geometric Series: Recognize that the series forms a geometric series with the first term $a_1 = \frac{5}{10}$ and a common ratio $r = \frac{1}{10}$ .
Use Infinite Series 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 $a_1 = \frac{5}{10}$ and $r = \frac{1}{10}$ into the formula. $\newline$ $S = \frac{(5/10)}{(1 - 1/10)}$
Simplify Expression: Simplify the expression. $\newline$ $S = \frac{5}{10} / \frac{9}{10}$ $\newline$ $S = \frac{5}{10} \times \frac{10}{9}$ $\newline$ $S = \frac{5}{9}$