Key Concepts and Formulas

• Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The sum of the first $N$ terms of a GP is given by $S_N = \frac{a(1 - r^N)}{1 - r}$, where $a$ is the first term.
• Transformation of Repeating Decimals: A repeating decimal of the form $0.\underbrace{d d \dots d}_{n \text{ times}}$ can be written as $\frac{d}{9}(1 - 10^{-n})$.
• Summation Properties: The sum of a constant $c$ over $N$ terms is $Nc$. The sum of terms can be separated: $\sum (a_n - b_n) = \sum a_n - \sum b_n$.

Step-by-Step Solution

Step 1: Express the $n$-th term of the sequence in a general form. The given sequence is $0.7, 0.77, 0.777, \dots$. Let $T_n$ be the $n$-th term. $T_n = 0.\underbrace{77\dots7}_{n \text{ times}}$ We can rewrite this as: $T_n = 7 \times 0.\underbrace{11\dots1}_{n \text{ times}}$

Step 2: Transform the term $0.\underbrace{11\dots1}_{n \text{ times}}$ into a more usable form. We use the formula for repeating decimals: $0.\underbrace{11\dots1}_{n \text{ times}} = \frac{1}{9}(1 - 10^{-n})$. Therefore, the $n$-th term becomes: $T_n = 7 \times \frac{1}{9}(1 - 10^{-n})$ $T_n = \frac{7}{9}(1 - 10^{-n})$

Step 3: Write the sum of the first 20 terms. We need to find $S_{20} = \sum_{n=1}^{20} T_n$. Substituting the transformed expression for $T_n$: $S_{20} = \sum_{n=1}^{20} \frac{7}{9}(1 - 10^{-n})$

Step 4: Separate the summation into constant and GP parts. We can factor out the constant $\frac{7}{9}$ and split the summation: $S_{20} = \frac{7}{9} \left( \sum_{n=1}^{20} 1 - \sum_{n=1}^{20} 10^{-n} \right)$

Step 5: Calculate the sum of the constant terms. The first part of the summation is $\sum_{n=1}^{20} 1$, which is the sum of 1 added 20 times. $\sum_{n=1}^{20} 1 = 1 \times 20 = 20$

Step 6: Identify and calculate the sum of the Geometric Progression. The second part of the summation is $\sum_{n=1}^{20} 10^{-n}$. This is a geometric progression: $10^{-1} + 10^{-2} + 10^{-3} + \dots + 10^{-20}$ Here, the first term ($a$) is $10^{-1} = \frac{1}{10}$. The common ratio ($r$) is $\frac{10^{-2}}{10^{-1}} = \frac{1}{10}$. The number of terms ($N$) is 20. Using the GP sum formula $S_N = \frac{a(1 - r^N)}{1 - r}$: $G_{20} = \frac{\frac{1}{10} \left( 1 - \left(\frac{1}{10}\right)^{20} \right)}{1 - \frac{1}{10}}$ $G_{20} = \frac{\frac{1}{10} (1 - 10^{-20})}{\frac{9}{10}}$ $G_{20} = \frac{1}{9}(1 - 10^{-20})$

Step 7: Substitute the calculated sums back into the expression for $S_{20}$. $S_{20} = \frac{7}{9} \left( 20 - G_{20} \right)$ $S_{20} = \frac{7}{9} \left( 20 - \frac{1}{9}(1 - 10^{-20}) \right)$

Step 8: Simplify the expression. Distribute the $\frac{1}{9}$ inside the parenthesis: $S_{20} = \frac{7}{9} \left( 20 - \frac{1}{9} + \frac{1}{9} 10^{-20} \right)$ Combine the constant terms $20 - \frac{1}{9}$: $20 - \frac{1}{9} = \frac{20 \times 9}{9} - \frac{1}{9} = \frac{180 - 1}{9} = \frac{179}{9}$ Substitute this back into the expression for $S_{20}$: $S_{20} = \frac{7}{9} \left( \frac{179}{9} + \frac{1}{9} 10^{-20} \right)$ Factor out $\frac{1}{9}$ from the terms inside the parenthesis: $S_{20} = \frac{7}{9} \times \frac{1}{9} \left( 179 + 10^{-20} \right)$ $S_{20} = \frac{7}{81} \left( 179 + 10^{-20} \right)$

This result matches option (A).

Common Mistakes & Tips

• Transformation Accuracy: Ensure the transformation of the repeating decimal $0.ddd\dots$ to $\frac{d}{9}(1 - 10^{-n})$ is correctly applied. A common error is using $\frac{d}{9}$ without the $(1 - 10^{-n})$ term.
• Sign Errors: Be meticulous with signs when subtracting the GP sum, especially after distributing the constant factor.
• GP Formula Application: Double-check the first term ($a$), common ratio ($r$), and number of terms ($N$) when applying the GP sum formula.

Summary

The problem involves summing a sequence of repeating decimals. The key strategy is to transform each term into a form that separates into a constant part and a geometric progression. By applying the formula for the sum of a geometric progression and carefully simplifying the resulting expression, we arrive at the sum of the first 20 terms. The transformation $0.\underbrace{77\dots7}_{n \text{ times}} = \frac{7}{9}(1 - 10^{-n})$ is crucial for breaking down the problem.

The final answer is \boxed{\text{{7 \over 81}}\left( {179 - {{10}^{ - 20}}} \right)}.