Write the repeating decimal as a fraction. .212212212 ______

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Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂212＂ repeat indefinitely.Express as Sum: Express the repeating decimal as a sum of its parts: 0.212212212... = 0.212 + 0.000212 + 0.000000212 + ...Convert to Fraction: Convert each part into a fraction: 0.212 = 212/1000, and notice that 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 = 212/1000 and the common ratio r = 1/1000.Use Sum Formula: Use the formula for the sum of an infinite geometric series, S = a_1 / (1 - r), where S is the sum, a_1 is the first term, and r is the common ratio.Substitute Values: Substitute the values into the formula: S = (212/1000) / (1 - 1/1000).Simplify Denominator: Simplify the denominator: 1 - 1/1000 = 999/1000.Calculate Sum: Now, calculate the sum: S = (212/1000) / (999/1000).Multiply by Reciprocal: Multiply by the reciprocal of the denominator: S = (212/1000) * (1000/999).Simplify Fraction: Simplify the fraction by multiplying the numerators and denominators: S = 212/999.Check for any possible simplification of the fraction 212/999. Since 212 and 999 have no common factors other than 1, the fraction is already in its simplest form.

Write the repeating decimal as a fraction. $\newline$ $.212212212$

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Write the repeating decimal as a fraction. $\newline$ $.212212212$