The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, .. , is (1) `7/9(99-10^(-20))` (2) `7/(81)(179+10^(-20))` (3) `7/9(99+10^(-20))` (3) `7/(81)(179-10^(-20))`

Options

A. `7/81(179-10)^(20))`
B. `7/9(99 -10^(20))`
C. `7/81(179 +10^(-20))`
D. `7/9(99+10^(-20))`

Explanation

To find the sum of the first 20 terms of the sequence 0.7, 0.77, 0.777, ..., we can follow these steps:

### Step 1: Identify the sequence
The sequence can be expressed as:
- \( a_1 = 0.7 \)
- \( a_2 = 0.77 \)
- \( a_3 = 0.777 \)
- ...
- \( a_n = 0.7 + 0.07 + 0.007 + ... + \frac{7}{10^n} \)

### Step 2: Rewrite the terms
Each term can be rewritten as:
\[ a_n = 0.7 + 0.07 + 0.007 + ... + \frac{7}{10^n} \]
This can be factored as:
\[ a_n = 0.7 \left(1 + 0.1 + 0.01 + ... + \frac{1}{10^{n-1}}\right) \]

### Step 3: Recognize the geometric series
The series inside the parentheses is a geometric series with:
- First term \( a = 1 \)
- Common ratio \( r = 0.1 \)
- Number of terms \( n \)

The sum \( S_n \) of the first \( n \) terms of a geometric series is given by:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Substituting the values:
\[ S_n = \frac{1(1 - (0.1)^n)}{1 - 0.1} = \frac{1 - (0.1)^n}{0.9} \]

### Step 4: Substitute back into the expression for \( a_n \)
Now substituting back into the expression for \( a_n \):
\[ a_n = 0.7 \cdot \frac{1 - (0.1)^n}{0.9} = \frac{7}{9} (1 - (0.1)^n) \]

### Step 5: Find the sum of the first 20 terms
Now, we need to find the sum of the first 20 terms:
\[ S_{20} = a_1 + a_2 + a_3 + ... + a_{20} \]
Using our expression for \( a_n \):
\[ S_{20} = \sum_{n=1}^{20} a_n = \sum_{n=1}^{20} \frac{7}{9} (1 - (0.1)^n) \]
This can be simplified:
\[ S_{20} = \frac{7}{9} \left(20 - \sum_{n=1}^{20} (0.1)^n\right) \]

### Step 6: Calculate the sum of the geometric series
The sum \( \sum_{n=1}^{20} (0.1)^n \) is also a geometric series with:
- First term \( a = 0.1 \)
- Common ratio \( r = 0.1 \)
- Number of terms \( n = 20 \)

Using the formula for the sum:
\[ \sum_{n=1}^{20} (0.1)^n = \frac{0.1(1 - (0.1)^{20})}{1 - 0.1} = \frac{0.1(1 - 10^{-20})}{0.9} = \frac{1 - 10^{-20}}{9} \]

### Step 7: Substitute back into the expression for \( S_{20} \)
Substituting this back into our expression for \( S_{20} \):
\[ S_{20} = \frac{7}{9} \left(20 - \frac{1 - 10^{-20}}{9}\right) \]
\[ = \frac{7}{9} \left(20 - \frac{1}{9} + \frac{10^{-20}}{9}\right) \]
\[ = \frac{7}{9} \left(\frac{180 - 1 + 10^{-20}}{9}\right) \]
\[ = \frac{7}{81} (179 + 10^{-20}) \]

### Final Answer
Thus, the sum of the first 20 terms of the sequence is:
\[ \frac{7}{81}(179 + 10^{-20}) \]