The value of `1 + (11)/(10) + (11)/(100) + (111)/(1000) + (111)/(10000)` is

Options

A. `3.3221`
B. `2.432`
C. `2.3321`
D. `2.245`

Answer

To find the value of the expression \(1 + \frac{11}{10} + \frac{11}{100} + \frac{111}{1000} + \frac{111}{10000}\), we will calculate each term step by step and then sum them up.

### Step 1: Calculate each term

1. The first term is simply:
\[ 1 = 1.0000 \]

2. The second term is:
\[ \frac{11}{10} = 1.1 = 1.1000 \]

3. The third term is:
\[ \frac{11}{100} = 0.11 = 0.1100 \]

4. The fourth term is:
\[ \frac{111}{1000} = 0.111 = 0.1110 \]

5. The fifth term is:
\[ \frac{111}{10000} = 0.0111 = 0.0111 \]

### Step 2: Align the decimals and sum the values

Now we will add all these values together, aligning them by their decimal points:

```
1.0000
+ 1.1000
+ 0.1100
+ 0.1110
+ 0.0111
___________
```

### Step 3: Add the numbers column by column

1. Start from the rightmost column:
- \(0 + 0 + 0 + 0 + 1 = 1\)

2. Move to the next column:
- \(0 + 0 + 1 + 1 + 1 = 3\)

3. Next column:
- \(0 + 1 + 1 + 1 + 0 = 3\)

4. Next column:
- \(1 + 1 + 0 + 0 + 0 = 2\)

5. Finally, the leftmost column:
- \(1 + 0 + 0 + 0 + 0 = 1\)

### Step 4: Write down the final sum

Putting it all together, we have:
\[ 2.3331 \]

### Conclusion

Thus, the value of the expression \(1 + \frac{11}{10} + \frac{11}{100} + \frac{111}{1000} + \frac{111}{10000}\) is:
\[ \boxed{2.3331} \]