A) \[\frac{5}{9}\left[ n-\frac{2}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\]
B) \[\frac{1}{9}\left[ 5-\frac{2}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\]
C) \[\frac{1}{9}\left[ n-\frac{5}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\]
D) \[\frac{5}{9}\left[ n-\frac{1}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\]
Correct Answer: D
Solution :
| [d] Given \[0.5+0.55+0.555+.....\text{ }to\,\,n\] |
| \[=5[0.1+0.11+0.111+.......to\,\,n\,\,terms]\] |
| \[=\frac{5}{9}[0.9+0.99+0.999+.......to\,\,n\,\,terms]\] |
| \[=\frac{5}{9}\left[ \frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+......\,\,to\,n\,terms \right]\] |
| \[=\frac{5}{9}\left[ \begin{align} & \left( 1-\frac{1}{10} \right)+\left( 1-\frac{1}{100} \right)+\left( 1-\frac{1}{1000} \right)+.... \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,to\,\,n\,\,terms \\ \end{align} \right]\] |
| \[=\frac{5}{9}\left[ \begin{align} & \left( 1-\frac{1}{10} \right)+\left( 1-\frac{1}{{{10}^{2}}} \right)+\left( 1-\frac{1}{{{10}^{3}}} \right)+.... \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( 1-\frac{1}{{{10}^{n}}} \right) \\ \end{align} \right]\] |
| \[=\frac{5}{9}\left[ n-\left( \frac{1}{10}+\frac{1}{{{10}^{2}}}+....\frac{1}{{{10}^{n}}} \right) \right]\] |
| \[=\frac{5}{9}\left[ n-\frac{1}{10}\frac{\left\{ 1-{{\left( \frac{1}{10} \right)}^{n}} \right\}}{\left( 1-\frac{1}{10} \right)} \right]\] |
| \[=\frac{5}{9}\left[ n-\frac{1}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] |
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