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Question
Find the sum of the following series:
0.5 + 0.55 + 0.555 + ... to n terms.
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Solution
We have,
0.5 + 0.55 + 0.555 + ... n terms
\[S_n\] = 5 [0.1 + 0.11+0.111 + ... n terms]
\[= \frac{5}{9}\left( 0 . 9 + 0 . 99 + 0 . 999 + . . . +\text { to n terms } \right)\]
\[ = \frac{5}{9}\left\{ \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + . . . \text { n terms } \right\}\]
\[ = \frac{5}{9}\left\{ \left( 1 - \frac{1}{10} \right) + \left( 1 - \frac{1}{100} \right) + \left( 1 - \frac{1}{1000} \right) + . . . \text { n terms } \right\} \]
\[ = \frac{5}{9}\left\{ n - \left( \frac{1}{10} + \frac{1}{{10}^2} + \frac{1}{{10}^3} + . . . \text { n terms } \right) \right\} \]
\[ = \frac{5}{9}\left\{ n - \frac{1}{10}\frac{\left( 1 - \left( \frac{1}{10} \right)^n \right)}{\left( 1 - \frac{1}{10} \right)} \right\}\]
\[ = \frac{5}{9}\left\{ n - \frac{1}{9}\left( 1 - \frac{1}{{10}^n} \right) \right\}\]
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