Question

The sum of the first 20 terms of the sequence 0, 7, 0.77, 0.777,...., is
(IIT JEE Main 2013)

• 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 D $\frac{7}{9}(99+{10}^{-20}$)

We have,
0.7 + 0.77 + 0.777 + ... upto 20 terms
7(0.1 + 0.11 + 0111+......upto 20 terms)
$\Rightarrow 7(\frac{1}{10}+\frac{11}{{10}^2}+\frac{111}{{10}^3}+...upto20terms)$

$\Rightarrow \frac{7}{9}(\frac{9}{10}+\frac{99}{{10}^2}+\frac{999}{{10}^3}+...upto20terms)$

$\Rightarrow \frac{7}{9}\left(\begin{array}{c}\left(\begin{array}{c}1-\frac{1}{10}\end{array}\right)+\left(\begin{array}{c}1-\frac{1}{100}\end{array}\right)+\left(\begin{array}{c}1-\frac{1}{1000}\end{array}\right)+....upto20terms\end{array}\right)$

$\Rightarrow \frac{7}{9}\left(\begin{array}{c}20-\left(\begin{array}{c}\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+....upto20terms\end{array}\right)\end{array}\right)$

$\Rightarrow \frac{7}{9}\left(\begin{array}{c}20-\left(\begin{array}{c}\frac{1}{10}\left(\begin{array}{c}\frac{1-{\left(\begin{array}{c}\frac{1}{10}\end{array}\right)}^{20}}{1-\frac{1}{10}}\end{array}\right)\end{array}\right)\end{array}\right)$

$\Rightarrow \frac{7}{9}\left(\begin{array}{c}20-\frac{1}{9}\left(\begin{array}{c}1-{\left(\begin{array}{c}\frac{1}{10}\end{array}\right)}^{20}\end{array}\right)\end{array}\right)$

$\Rightarrow \frac{7}{81}\left(\begin{array}{c}179+{\left(\begin{array}{c}\frac{1}{10}\end{array}\right)}^{20}\end{array}\right)$