Find the sum of n terms of 0.8 + 0.88 + 0.888 + …

Options

A. `8/9 [1/9 {1-(0.1)^(n)}]`
B. `8/9 [1/(10) {1-(0.1)^(n)}]`
C. `8/9 [n-1/(9) {1-(0.1)^(n)}]`
D. none of these

Answer

To find the sum of the first n terms of the series 0.8 + 0.88 + 0.888 + ..., we can follow these steps:

### Step 1: Identify the Pattern
The series can be expressed as:
- First term: 0.8
- Second term: 0.88
- Third term: 0.888

We can see that each term can be rewritten in a more manageable form.

### Step 2: Rewrite the Terms
Notice that:
- 0.8 = 8/10
- 0.88 = 88/100 = 8/10 + 8/100
- 0.888 = 888/1000 = 8/10 + 8/100 + 8/1000

Thus, we can express the nth term as:
\[ a_n = \frac{8}{10} + \frac{8}{100} + \frac{8}{1000} + ... + \frac{8}{10^n} \]

### Step 3: Factor Out the Common Factor
We can factor out 8 from each term:
\[ S_n = 8 \left( \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + ... + \frac{1}{10^n} \right) \]

### Step 4: Recognize the Series as a Geometric Series
The series inside the parentheses is a geometric series where:
- First term \( a = \frac{1}{10} \)
- Common ratio \( r = \frac{1}{10} \)
- Number of terms \( n \)

### Step 5: Use the Formula for the Sum of a Geometric Series
The sum \( S \) of the first n terms of a geometric series can be calculated using the formula:
\[ S = a \frac{1 - r^n}{1 - r} \]

Substituting the values:
\[ S = \frac{1}{10} \cdot \frac{1 - \left(\frac{1}{10}\right)^n}{1 - \frac{1}{10}} \]
\[ S = \frac{1}{10} \cdot \frac{1 - \frac{1}{10^n}}{\frac{9}{10}} \]
\[ S = \frac{1}{10} \cdot \frac{10}{9} \left(1 - \frac{1}{10^n}\right) \]
\[ S = \frac{1}{9} \left(1 - \frac{1}{10^n}\right) \]

### Step 6: Substitute Back to Find the Total Sum
Now, substituting back into our expression for \( S_n \):
\[ S_n = 8 \cdot \frac{1}{9} \left(1 - \frac{1}{10^n}\right) \]
\[ S_n = \frac{8}{9} \left(1 - \frac{1}{10^n}\right) \]

### Final Answer
Thus, the sum of the first n terms of the series is:
\[ S_n = \frac{8}{9} \left(1 - \frac{1}{10^n}\right) \]

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