Each term can be written as \(0.7, 0.77, 0.777, \ldots = \frac{7}{9}(0.9, 0.99, 0.999, \ldots) = \frac{7}{9}(1-10^{-1}, 1-10^{-2}, 1-10^{-3}, \ldots)\). The sum of first 20 terms is \(S = \frac{7}{9}\sum_{k=1}^{20}(1-10^{-k}) = \frac{7}{9}\left[20 - \sum_{k=1}^{20}10^{-k}\right] = \frac{7}{9}\left[20 - \frac{0.1(1-10^{-20})}{1-0.1}\right] = \frac{7}{9}\left[20 - \frac{1-10^{-20}}{9}\right] = \frac{7}{9} \cdot \frac{180 - 1 + 10^{-20}}{9} = \frac{7}{81}(179 + 10^{-20})\).