The sum of the square of three consecutive odd numbers is 2195. Find the numbers.

Options

A. 21, 23, 25
B. 23, 25, 27
C. 25, 27, 29
D. 27, 29, 31

Answer

To solve the problem of finding three consecutive odd numbers whose squares sum up to 2195, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Odd Numbers**:
Let the three consecutive odd numbers be represented as:
- First number: \( x - 2 \)
- Second number: \( x \)
- Third number: \( x + 2 \)

2. **Set Up the Equation**:
According to the problem, the sum of the squares of these numbers is equal to 2195. Therefore, we can write the equation:
\[ (x - 2)^2 + x^2 + (x + 2)^2 = 2195 \]

3. **Expand the Squares**:
Now, we will expand each term in the equation:
- \( (x - 2)^2 = x^2 - 4x + 4 \)
- \( x^2 = x^2 \)
- \( (x + 2)^2 = x^2 + 4x + 4 \)

Combining these, we get:
\[ (x^2 - 4x + 4) + x^2 + (x^2 + 4x + 4) = 2195 \]

4. **Combine Like Terms**:
Now, combine all the terms:
\[ x^2 - 4x + 4 + x^2 + x^2 + 4x + 4 = 2195 \]
This simplifies to:
\[ 3x^2 + 8 = 2195 \]

5. **Isolate the Quadratic Term**:
Next, subtract 8 from both sides:
\[ 3x^2 = 2195 - 8 \]
\[ 3x^2 = 2187 \]

6. **Solve for \( x^2 \)**:
Divide both sides by 3:
\[ x^2 = \frac{2187}{3} \]
\[ x^2 = 729 \]

7. **Find \( x \)**:
Now, take the square root of both sides:
\[ x = \sqrt{729} = 27 \]

8. **Determine the Odd Numbers**:
Finally, substitute \( x \) back to find the three consecutive odd numbers:
- First number: \( x - 2 = 27 - 2 = 25 \)
- Second number: \( x = 27 \)
- Third number: \( x + 2 = 27 + 2 = 29 \)

Therefore, the three consecutive odd numbers are **25, 27, and 29**.

### Final Answer:
The three consecutive odd numbers are **25, 27, and 29**.