Write the repeating decimal as a fraction. .22222222 ______

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating decimal is .22222222..., where the digit 2 repeats indefinitely.Express as Infinite Sum: Express the repeating decimal as an infinite sum of its terms. The repeating decimal .22222222... can be written as 0.2 + 0.02 + 0.002 + ...Convert to Fractions: Convert each term of the sum into a fraction. This gives us 2/10 + 2/100 + 2/1000 + ...Recognize Geometric Series: Recognize that the series 2/10 + 2/100 + 2/1000 + ... is a geometric series with the first term a_1 = 2/10 and a common ratio r = 1/10.Use Sum Formula: Use the formula for the sum of an infinite geometric series, which is a_1 / (1 - r), to write the repeating decimal as a fraction. Substitute a_1 = 2/10 and r = 1/10 into the formula.Perform Calculation: Perform the calculation: (2/10) / (1 - 1/10) = (2/10) / (9/10) = 2/10 * 10/9 = 2/9.Conclude as Fraction: Conclude that the repeating decimal .22222222... is equal to the fraction 2/9.

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Write the repeating decimal as a fraction. $\newline$ $.22222222$

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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

.22222222

Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating decimal is $0.22222222\ldots$ , where the digit $2$ repeats indefinitely.
Express as Infinite Sum: Express the repeating decimal as an infinite sum of its terms. The repeating decimal $0.22222222\ldots$ can be written as $0.2 + 0.02 + 0.002 + \ldots$
Convert to Fractions: Convert each term of the sum into a fraction. This gives us $\frac{2}{10} + \frac{2}{100} + \frac{2}{1000} + \ldots$
Recognize Geometric Series: Recognize that the series $\frac{2}{10} + \frac{2}{100} + \frac{2}{1000} + \ldots$ is a geometric series with the first term $a_1 = \frac{2}{10}$ and a common ratio $r = \frac{1}{10}$ .
Use Sum Formula: Use the formula for the sum of an infinite geometric series, which is $\frac{a_1}{1 - r}$ , to write the repeating decimal as a fraction. Substitute $a_1 = \frac{2}{10}$ and $r = \frac{1}{10}$ into the formula.
Perform Calculation: Perform the calculation: $(\frac{2}{10}) / (1 - \frac{1}{10}) = (\frac{2}{10}) / (\frac{9}{10}) = \frac{2}{10} \times \frac{10}{9} = \frac{2}{9}$ .
Conclude as Fraction: Conclude that the repeating decimal $0.22222222\ldots$ is equal to the fraction $\frac{2}{9}$ .