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Question
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
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Solution
\[ 3 . 5\overline 2\]
\[\text { Let } S = 3 . 5\overline 2\]
\[ \Rightarrow S = 3 . 5 + 0 . 02 + 0 . 002 + 0002 + 0 . 00002 + . . . \infty \]
\[ \Rightarrow S = 3 . 5 + 0 . 02\left( 1 + {10}^{- 1} + {10}^{- 2} + {10}^{- 3} + {10}^{- 4} + . . . \infty \right)\]
\[\text { It is a G . P } . \]
\[ \therefore S = 3 . 5 + 0 . 02\left( \frac{1}{1 - {10}^{- 1}} \right)\]
\[ \Rightarrow S = 3 . 5 + \frac{0 . 2}{9}\]
\[ \Rightarrow S = \frac{317}{90}\]
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