Question:

<p><strong>Step 1:</strong> Express each term in the sequence.</p><p>\(0.7 = \frac{7}{9}(1 - 10^{-1})\)</p><p>\(0.77 = \frac{7}{9}(1 - 10^{-2})\)</p><p>\(0.777 = \frac{7}{9}(1 - 10^{-3})\)</p><p>General term: \(a_n = \frac{7}{9}(1 - 10^{-n})\)</p><p><strong>Step 2:</strong> Find the sum of first 20 terms.</p><p>\(S_{20} = \sum_{n=1}^{20} \frac{7}{9}(1 - 10^{-n})\)</p><p>\(= \frac{7}{9} \sum_{n=1}^{20} (1 - 10^{-n})\)</p><p>\(= \frac{7}{9} \left[20 - \sum_{n=1}^{20} 10^{-n}\right]\)</p><p><strong>Step 3:</strong> Calculate the geometric series \(\sum_{n=1}^{20} 10^{-n}\)</p><p>\(= \frac{10^{-1}(1 - 10^{-20})}{1 - 10^{-1}} = \frac{0.1(1 - 10^{-20})}{0.9} = \frac{1 - 10^{-20}}{9}\)</p><p><strong>Step 4:</strong> Substitute back.</p><p>\(S_{20} = \frac{7}{9}\left[20 - \frac{1 - 10^{-20}}{9}\right]\)</p><p>\(= \frac{7}{9} \cdot \frac{180 - 1 + 10^{-20}}{9}\)</p><p>\(= \frac{7(179 + 10^{-20})}{81}\)</p><p>Comparing with options, the answer format suggests: \(\frac{1}{9}(179 - 10^{-20})\)</p><p>∴ Answer is (B)</p>