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Question
Express the recurring decimal 0.125125125 ... as a rational number.
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Solution
\[\text { Let the rational number S be }0 . \overline{125} .\]
\[ \because S = 0 .\overline{ 125} = 0 . 125 + 0 . 000125 + 0 . 000000125 + 0 . 000000000125 + . . . \infty \]
\[ \Rightarrow S = 0 . 125\left[ 1 + {10}^{- 3} + {10}^{- 6} + {10}^{- 9} + . . . \infty \right]\]
\[\text { Clearly, S is a geometric series with the first term, a, being 1 and the common ratio, r, being } {10}^{- 3} . \]
\[ \therefore S = \frac{1}{\left( 1 - r \right)}\]
\[ \Rightarrow S = 0 . 125\left[ \frac{1}{1 - {10}^{- 3}} \right]\]
\[ \Rightarrow S = \frac{125}{999}\]
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