Sum to 25 terms of the series 0.5+0.55 +0.555+…is :

Options

A. `(5)/(81)(224-10^(-25))`
B. `(5)/(9)(224-10^(-25))`
C. `(5)/(81)(224-10^(-24))`
D. none of these

Answer

To find the sum of the first 25 terms of the series 0.5 + 0.55 + 0.555 + ..., we can follow these steps:

### Step 1: Identify the Pattern
The series can be expressed as:
- 0.5 = 5/10
- 0.55 = 55/100 = 5.5/10
- 0.555 = 555/1000 = 5.55/10

We can see that each term can be written as:
\[ a_n = \frac{5}{10} + \frac{5}{100} + \frac{5}{1000} + ... \]

### Step 2: Factor Out the Common Term
We can factor out 5 from the series:
\[ S = 5 \left( \frac{1}{10} + \frac{11}{100} + \frac{111}{1000} + ... \right) \]

### Step 3: Rewrite the Series
Notice that:
- The first term is \( 0.1 \)
- The second term is \( 0.11 \)
- The third term is \( 0.111 \)

This can be expressed as:
\[ S = 5 \left( 0.1 + 0.11 + 0.111 + ... \right) \]

### Step 4: Express in Terms of a Geometric Series
The series inside the parentheses can be rewritten as:
\[ S = 5 \left( \frac{1}{10} + \frac{11}{100} + \frac{111}{1000} + ... \right) \]

This can be simplified to:
\[ S = 5 \left( \frac{1}{10} + \frac{1}{10} \cdot \frac{1}{10} + \frac{1}{10} \cdot \frac{1}{100} + ... \right) \]

### Step 5: Identify the Geometric Series
The series \( 0.1, 0.11, 0.111, ... \) can be recognized as a geometric series with:
- First term \( a = \frac{1}{10} \)
- Common ratio \( r = \frac{1}{10} \)

### Step 6: Use the Formula for the Sum of a Geometric Series
The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]

For our series:
- \( n = 25 \)
- \( a = \frac{1}{10} \)
- \( r = \frac{1}{10} \)

Thus, the sum becomes:
\[ S_{25} = \frac{1/10 \cdot (1 - (1/10)^{25})}{1 - 1/10} \]
\[ S_{25} = \frac{1/10 \cdot (1 - (1/10)^{25})}{9/10} \]
\[ S_{25} = \frac{1 - (1/10)^{25}}{9} \]

### Step 7: Calculate the Total Sum
Now, substituting back into the equation for \( S \):
\[ S = 5 \cdot S_{25} = 5 \cdot \frac{1 - (1/10)^{25}}{9} \]
\[ S = \frac{5(1 - (1/10)^{25})}{9} \]

### Step 8: Final Calculation
Now we can calculate the final value:
\[ S = \frac{5(1 - 10^{-25})}{9} \]