If `1^(2) +2^(2) + …+ 9^(2)=285`, then the value of `(0.11)^(2) + (0.22)^(2) + …+ (0.99)^(2)` is

Options

A. 3.4485
B. 2.4485
C. 0.24485
D. 0.34485

Answer

To solve the problem step by step, we will calculate the value of \( (0.11)^2 + (0.22)^2 + \ldots + (0.99)^2 \) using the information given.

### Step 1: Understand the series
The series given is:
\[ (0.11)^2 + (0.22)^2 + (0.33)^2 + (0.44)^2 + (0.55)^2 + (0.66)^2 + (0.77)^2 + (0.88)^2 + (0.99)^2 \]
This can be rewritten in terms of fractions:
\[ \left(\frac{11}{100}\right)^2 + \left(\frac{22}{100}\right)^2 + \left(\frac{33}{100}\right)^2 + \ldots + \left(\frac{99}{100}\right)^2 \]

### Step 2: Factor out common terms
Notice that each term can be expressed as:
\[ \left(\frac{11k}{100}\right)^2 \quad \text{for } k = 1, 2, \ldots, 9 \]
Thus, we can factor out \( \left(\frac{11}{100}\right)^2 \):
\[ \left(\frac{11}{100}\right)^2 \left(1^2 + 2^2 + 3^2 + \ldots + 9^2\right) \]

### Step 3: Substitute the known sum
From the problem, we know that:
\[ 1^2 + 2^2 + 3^2 + \ldots + 9^2 = 285 \]
So we can substitute this into our expression:
\[ \left(\frac{11}{100}\right)^2 \times 285 \]

### Step 4: Calculate \( \left(\frac{11}{100}\right)^2 \)
Calculating \( \left(\frac{11}{100}\right)^2 \):
\[ \left(\frac{11}{100}\right)^2 = \frac{121}{10000} \]

### Step 5: Multiply by 285
Now we multiply:
\[ \frac{121}{10000} \times 285 \]

### Step 6: Simplify the multiplication
First, we can simplify:
\[ \frac{121 \times 285}{10000} \]

### Step 7: Calculate \( 121 \times 285 \)
Calculating \( 121 \times 285 \):
\[ 121 \times 285 = 34485 \]

### Step 8: Divide by 10000
Now we divide:
\[ \frac{34485}{10000} = 3.4485 \]

### Final Answer
Thus, the value of \( (0.11)^2 + (0.22)^2 + \ldots + (0.99)^2 \) is:
\[ \boxed{3.4485} \]