The sum of first 20 terms of the sequence 0.7,0.77,0.777,...... is 

$T_r=\mathrm{0.7777......7}$ (r terms)

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

$S_{20}=\sum _{r=1}^{20}{T}_r=\frac{7}{9}\left(20-\sum _{r=1}^{20}{10}^{-r}\right)=\frac{7}{81}\left(179+{10}^{-20}\right)$