Question

The sum of the first 20 terms of the sequence 0.7, 0.77, 0.777, ....., is ___.

• A. $\frac{7}{81}(179-{10}^{-20})$
• B. $\frac{7}{9}(99-{10}^{-20})$
• C. $\frac{7}{81}(179+{10}^{-20})$
• D. $\frac{7}{9}(99+{10}^{-20})$

Answer 1

The correct option is C $\frac{7}{81}(179+{10}^{-20})$

$S=0.7+0.77+0.777+\dots \dots \text{upto 20 terms}$
$=\frac{7}{10}+\frac{77}{{10}^2}+\frac{777}{{10}^3}+\dots \dots \text{upto 20 terms}$
$=7[\frac{1}{10}+\frac{11}{{10}^2}+\frac{111}{{10}^3}+\dots \dots \text{upto 20 terms}]$
$=\frac{7}{9}[(1-\frac{1}{10})+(1-\frac{1}{{10}^2})+(1-\frac{1}{{10}^3})+\dots \dots +\text{upto 20 terms}]$
$=\frac{7}{9}\left[\right(1+1+\dots \dots +\text{upto 20 terms})-(\frac{1}{10}+\frac{1}{{10}^2}+\frac{1}{{10}^3}+\dots \dots +\text{upto 20 terms})]$
$=\frac{7}{9}[20-\frac{\frac{1}{10}\{1-{\left(\frac{1}{10}\right)}^{20}\}}{1-\frac{1}{10}}]$
$=\frac{7}{9}[20-\frac{1}{9}\{1-{\left(\frac{1}{10}\right)}^{20}\}]$
$=\frac{7}{9}[\frac{179}{9}+\frac{1}{9}{\left(\frac{1}{10}\right)}^{20}]=\frac{7}{81}[179+(10{)}^{-20}]$