`1+1/10+2/(10^2)+2/(10^3)+2/(10^4)+....?`

Options

A. `1.1bar2`
B. 0.121
C. 1.21
D. `1.bar(12)`

Answer

To solve the given series \( S = 1 + \frac{1}{10} + \frac{2}{10^2} + \frac{2}{10^3} + \frac{2}{10^4} + \ldots \), we can break it down into manageable parts.

### Step-by-Step Solution

1. **Identify the Series**:
The series can be rewritten as:
\[ S = 1 + \frac{1}{10} + \sum_{n=2}^{\infty} \frac{2}{10^n} \]

2. **Calculate the First Two Terms**:
The first two terms of the series are:
\[ 1 + \frac{1}{10} = 1.1 \]

3. **Identify the Infinite Series**:
The remaining part of the series is:
\[ \sum_{n=2}^{\infty} \frac{2}{10^n} \]
This is a geometric series where the first term \( a = \frac{2}{10^2} = \frac{2}{100} = 0.02 \) and the common ratio \( r = \frac{1}{10} \).

4. **Sum the Infinite Series**:
The sum of an infinite geometric series is given by the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{0.02}{1 - \frac{1}{10}} = \frac{0.02}{\frac{9}{10}} = 0.02 \times \frac{10}{9} = \frac{0.2}{9} \approx 0.0222\ldots \]

5. **Combine the Results**:
Now we add this sum to the first two terms:
\[ S = 1.1 + 0.0222\ldots \approx 1.1222\ldots \]

6. **Final Result**:
The final result can be expressed as:
\[ S \approx 1.12\overline{2} \]
This indicates that the digit '2' repeats infinitely.

### Conclusion
The value of the series is approximately \( 1.12\overline{2} \), which corresponds to option 1 in the given choices.