Question

The sum of 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 $\frac{7}{81}[179+{10}^{-20}]$

$S=0.7+0.77+0.777+...\text{upto}20\text{terms}$
$\Rightarrow S=\frac{7}{9}[0.9+0.99+0.999+...\text{upto}20\text{terms}$
$\Rightarrow \frac{9S}{7}=(1-0.1)+(1-0.01)+(1-0.001)+...$
$\Rightarrow \frac{9S}{7}=20-[\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+...\text{upto}20\text{terms}]$
$=20-\frac{1}{10}\left(\frac{1-{\left(\frac{1}{10}\right)}^{20}}{1-\frac{1}{10}}\right)$
$\Rightarrow \frac{9S}{7}=20-\frac{1-{10}^{-20}}{9}$
$\Rightarrow S=\frac{7}{81}[179+{10}^{-20}]$