Question

The sum of first $20$ terms of the sequence $0.7,0.77,0.777,$$\cdots$ , is

• A. $\frac{7}{9}(99-{10}^{-20})$
• B. $\frac{7}{81}(179+{10}^{-20})$
• C. $\frac{7}{9}(99+{10}^{-20})$
• D. $\frac{7}{81}(179-{10}^{-20})$

Answer 1

Answer: (B)

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

$t_r=0.777\dots$.. $r$ times

$=7({10}^{-1}+{10}^{-2}+{10}^{-3}+\dots ..+{10}^{-r})$

$=\frac{7}{9}(1-{10}^{-r})$

$S_{20}={\sum }_{r=1}^{20}{t}_r$

$=\frac{7}{9}(20-{\sum }_{r=1}^{20}{10}^{-r})$

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

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