Question

The sum of $10$ terms of the series $.7+.77+.777+.....$ is.

• A. $\frac{7}{9}(89+\frac{1}{{10}^{10}})$
• B. $\frac{7}{81}(89+\frac{1}{{10}^9})$
• C. $\frac{7}{81}(89+\frac{1}{{10}^{10}})$
• D. none of these

Answer 1

The correct option is C $\frac{7}{81}(89+\frac{1}{{10}^{10}})$

Sum $=.7+.77+.777...(10$ terms$)$

$=7(0.1+0.11+\mathrm{0.111....}({10}^{th}$ term$)$

$=\frac{7}{9}(0.9+0.99+0.999...({10}^{th}$ term$)$

$=\frac{7}{9}(1-0.1+1-0.01+1-0.001....({10}^{th}$ term$)$

$=\frac{7}{9}[10-\frac{0.1[1-(0.1{)}^{10}]}{1-0.1}]$

$=\frac{7}{9}[10-\frac{1-(0.1{)}^{10}}{9}]$

$=\frac{7}{9}\left[\frac{90-1+(0.1{)}^{10}}{9}\right]$

$=\frac{7}{81}[89+\frac{1}{{10}^{10}}]$

Then answer is $C$.