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Express the decimal x = 0.51515151… as a quotient of two integers

Let

    \[ x = 0.51515151\ldots. \]

Express x as an infinite series, find the sum, and express x as a quotient of two integers.


We have

    \begin{align*}  x = 0.515151\ldots && \implies && x &= \sum_{k=1}^{\infty} \frac{51}{100^k} \\[9pt]  &&&&&= 51 \sum_{k=0}^{\infty} \left( \frac{1}{100} \right)^k - 51 \\[9pt]  &&&&&= 51 \left( \frac{1}{1-\frac{1}{100}} \right) - 51 \\[9pt]  &&&&&= \frac{51}{99}. \end{align*}

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