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 $\frac{7}{81}(179+{10}^{-20})$

Given sequence can be written as
$\frac{7}{10}+\frac{77}{100}+\frac{777}{{10}^3}+...+$ up to 20 terms
$=7[\frac{1}{10}+\frac{11}{100}+\frac{111}{{10}^3}+...+upto20terms]$
Multiply and divide by 9
$\frac{7}{9}[(1-\frac{1}{10}+(1-{\frac{1}{10}}^2)+(1-{\frac{1}{10}}^3\left)\right)+.....upto20terms]=\frac{7}{9}[20-\frac{\frac{1}{10}(1-(\frac{1}{10}{)}^{20})}{1-\frac{1}{10}}]=\frac{7}{9}[\frac{179}{9}+\frac{1}{9}(\frac{1}{10}{)}^{20}]=\frac{7}{81}[179+(10{)}^{-20}]$