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Question
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
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Solution
\[0 . \overline {231 }\]
\[\text { Let } S = 0 . \overline {231 }\]
\[ \Rightarrow S = 0 . 231 + 0 . 000231 + 0 . 000000231 + . . . \infty \]
\[ \Rightarrow S = 0 . 231\left( 1 + {10}^{- 3} + {10}^{- 6} + . . . \infty \right)\]
\[\text { It is a G . P } . \]
\[ \therefore S = 0 . 231\left( \frac{1}{1 - {10}^{- 3}} \right)\]
\[ \Rightarrow S = \frac{231}{999}\]
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