A) \[\frac{7}{81}(179-{{10}^{-20}})\]
B) \[\frac{7}{9}(99-{{10}^{-20}})\]
C) \[\frac{7}{81}(179+{{10}^{-20}})\]
D) \[\frac{7}{9}(99+{{10}^{-20}})\]
Correct Answer: C
Solution :
\[0.7+0.77+0.777+\ldots \ldots +0.777\ldots \text{ }7\] \[=\frac{7}{9}[0.9+0.99+0.999+...+0.999..9]\] \[=\frac{7}{9}[(1-0.1)+(1-0.01)+(1-0.001)+...+(1-0.000...1)]\] \[=\frac{7}{9}\left[ 20-\left( \frac{1}{10}+\frac{1}{{{10}^{2}}}+\frac{1}{{{10}^{3}}}+...+\frac{1}{{{10}^{20}}} \right) \right]\] \[=\frac{7}{9}\left[ 20-\frac{1}{10}.\frac{1-\frac{1}{{{10}^{20}}}}{1-\frac{1}{10}} \right]\] \[=\frac{7}{9}\left[ 20-\frac{1}{9}.\left( \frac{{{10}^{20}}-1}{{{10}^{20}}} \right) \right]\] \[=\frac{7}{81}\left[ 180-\left( 1-\frac{1}{{{10}^{20}}} \right) \right]=\frac{7}{81}[179+{{10}^{-20}}]\]
You need to login to perform this action.
You will be redirected in
3 sec
