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 solve the expression \(1 + \frac{11}{10} + \frac{11}{100} + \frac{111}{1000} + \frac{111}{10000}\), we will follow these steps:

### Step 1: Find a common denominator
The denominators in the expression are \(1\), \(10\), \(100\), \(1000\), and \(10000\). The least common multiple (LCM) of these denominators is \(10000\).

### Step 2: Rewrite each term with the common denominator
We will convert each term to have a denominator of \(10000\):

- The first term \(1\) can be rewritten as \(\frac{10000}{10000}\).
- The second term \(\frac{11}{10}\) can be rewritten as \(\frac{11 \times 1000}{10 \times 1000} = \frac{11000}{10000}\).
- The third term \(\frac{11}{100}\) can be rewritten as \(\frac{11 \times 100}{100 \times 100} = \frac{1100}{10000}\).
- The fourth term \(\frac{111}{1000}\) can be rewritten as \(\frac{111 \times 10}{1000 \times 10} = \frac{1110}{10000}\).
- The fifth term \(\frac{111}{10000}\) remains \(\frac{111}{10000}\).

Now we can rewrite the entire expression as:
\[ \frac{10000}{10000} + \frac{11000}{10000} + \frac{1100}{10000} + \frac{1110}{10000} + \frac{111}{10000} \]

### Step 3: Combine the numerators
Now, we can combine all the numerators over the common denominator:
\[ \frac{10000 + 11000 + 1100 + 1110 + 111}{10000} \]

Calculating the sum of the numerators:
\[ 10000 + 11000 = 21000 \]
\[ 21000 + 1100 = 22100 \]
\[ 22100 + 1110 = 23210 \]
\[ 23210 + 111 = 23321 \]

So, we have:
\[ \frac{23321}{10000} \]

### Step 4: Convert to decimal form
Now we can convert \(\frac{23321}{10000}\) to decimal form:
\[ 23321 \div 10000 = 2.3321 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{2.3321} \]