Mathematics - Sequence & Series Question with Solution | TestHub

Let, $S=0·7+0·77+0·777+.....$,

Here, number of terms is $n$.

$\Rightarrow S=\frac{7}{9}(0·9+0·99+0·999+....)$
$\Rightarrow S=\frac{7}{9}\left(\frac{9}{10}+\frac{99}{{10}^2}+\frac{999}{{10}^3}+...\right)$

$\Rightarrow S=\frac{7}{9}\left(\frac{10-1}{10}+\frac{{10}^2-1}{{10}^2}+\frac{{10}^3-1}{{10}^3}+...\right)$

$\Rightarrow S=\frac{7}{9}\left(1+1+1+1+1+...+1\right)-\left(\frac{1}{10}+\frac{1}{{10}^2}+\frac{1}{{10}^3}+...+\frac{1}{{10}^n}\right)$
$\Rightarrow S=\frac{7}{9}\left(n-\frac{1}{10}\times \frac{\left(1-\frac{1}{{10}^n}\right)}{1-\frac{1}{10}}\right)$

For the sum of $20$ terms, put $n=20$.

$\Rightarrow S=\frac{7}{9}\left(20-\frac{1}{10}\times \frac{\left(1-\frac{1}{10^{20}}\right)}{1-\frac{1}{10}}\right)$

$\text{⇒S}=\frac{7}{9}\left(20-\frac{1-10^{-20}}{9}\right)$

$\text{⇒S}=\frac{7}{81}\left(179+10^{-20}\right)$