The sum of `1/10+23/10^3+23/10^5+23/10^7+....` as a recurring decimal fraction is

Options

A. `0.1bar(23)`
B. `0.bar(123)`
C. `0.12bar(3)`
D. `0.1234`

Answer

To find the sum of the series \( \frac{1}{10} + \frac{23}{10^3} + \frac{23}{10^5} + \frac{23}{10^7} + \ldots \), we can break it down into manageable parts.

### Step 1: Separate the first term from the series

We can separate the first term \( \frac{1}{10} \) from the rest of the series:

\[ S = \frac{1}{10} + \left( \frac{23}{10^3} + \frac{23}{10^5} + \frac{23}{10^7} + \ldots \right) \]

### Step 2: Factor out the common term from the series

The remaining series can be factored. Notice that each term after the first can be expressed as \( \frac{23}{10^{2n+1}} \) for \( n = 1, 2, 3, \ldots \). Thus, we can factor out \( \frac{23}{10^3} \):

\[ S = \frac{1}{10} + \frac{23}{10^3} \left( 1 + \frac{1}{10^2} + \frac{1}{10^4} + \ldots \right) \]

### Step 3: Identify the geometric series

The series in the parentheses is a geometric series with the first term \( a = 1 \) and the common ratio \( r = \frac{1}{10^2} = \frac{1}{100} \). The sum of an infinite geometric series is given by the formula:

\[ \text{Sum} = \frac{a}{1 - r} \]

### Step 4: Calculate the sum of the geometric series

Substituting the values of \( a \) and \( r \):

\[ \text{Sum} = \frac{1}{1 - \frac{1}{100}} = \frac{1}{\frac{99}{100}} = \frac{100}{99} \]

### Step 5: Substitute back into the equation for \( S \)

Now we substitute this back into our equation for \( S \):

\[ S = \frac{1}{10} + \frac{23}{10^3} \cdot \frac{100}{99} \]

### Step 6: Simplify the second term

Calculating the second term:

\[ \frac{23 \cdot 100}{10^3 \cdot 99} = \frac{2300}{99000} = \frac{23}{990} \]

### Step 7: Combine the terms

Now we need to combine \( \frac{1}{10} \) and \( \frac{23}{990} \). To do this, we need a common denominator. The least common multiple of 10 and 990 is 990. Thus, we rewrite \( \frac{1}{10} \):

\[ \frac{1}{10} = \frac{99}{990} \]

Now we can combine:

\[ S = \frac{99}{990} + \frac{23}{990} = \frac{99 + 23}{990} = \frac{122}{990} \]

### Step 8: Simplify the fraction

Now we simplify \( \frac{122}{990} \):

\[ \frac{122 \div 2}{990 \div 2} = \frac{61}{495} \]

### Step 9: Convert to a recurring decimal

To express \( S \) as a recurring decimal, we can perform the division \( 61 \div 495 \):

\[ 61 \div 495 \approx 0.1232323\ldots \]

This can be expressed as \( 0.123\overline{23} \).

### Final Answer

Thus, the sum of the series \( \frac{1}{10} + \frac{23}{10^3} + \frac{23}{10^5} + \frac{23}{10^7} + \ldots \) as a recurring decimal fraction is:

\[ \boxed{0.123\overline{23}} \]