A) 1
B) \[\frac{1}{4}\]
C) \[\frac{1}{3}\]
D) \[\frac{3}{4}\]
Correct Answer: D
Solution :
\[\sum\limits_{n=1}^{\infty }{\frac{{{5}^{n}}-{{2}^{n}}}{{{10}^{n}}}=\sum\limits_{n=1}^{\infty }{\left( \frac{{{5}^{n}}}{{{10}^{n}}}-\frac{{{2}^{n}}}{{{10}^{n}}} \right)}}={{\sum\limits_{n=1}^{\infty }{\left( \frac{1}{2} \right)}}^{n}}-{{\left( \frac{1}{5} \right)}^{n}};\] The difference between two infinite geometric sequences. \[=\frac{\frac{1}{2}}{1-\frac{1}{2}}-\frac{\frac{1}{5}}{1-\frac{1}{5}}=1-\frac{1}{4}=\frac{3}{4}\]
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