Three number are in A.P. such that their sum is
24 and sum of their squares is 200 . The number are

Options

A. 2, 8, 14 `
B. `4, 8, 12 `
C. 6,8,10
D. 5, 8, 11

Answer

To solve the problem, we need to find three numbers in Arithmetic Progression (A.P.) such that their sum is 24 and the sum of their squares is 200.

### Step-by-Step Solution:

1. **Define the Numbers**:
Let the three numbers in A.P. be:
- First number: \( A - D \)
- Second number: \( A \)
- Third number: \( A + D \)

2. **Set Up the Sum Equation**:
According to the problem, the sum of these three numbers is 24:
\[ (A - D) + A + (A + D) = 24 \]
Simplifying this, we get:
\[ 3A = 24 \]
Therefore, solving for \( A \):
\[ A = \frac{24}{3} = 8 \]

3. **Set Up the Sum of Squares Equation**:
The sum of their squares is given as 200:
\[ (A - D)^2 + A^2 + (A + D)^2 = 200 \]
Expanding this, we have:
\[ (A^2 - 2AD + D^2) + A^2 + (A^2 + 2AD + D^2) = 200 \]
Combining like terms:
\[ 3A^2 + 2D^2 = 200 \]

4. **Substitute the Value of A**:
Now, substitute \( A = 8 \) into the equation:
\[ 3(8^2) + 2D^2 = 200 \]
Calculating \( 8^2 \):
\[ 3(64) + 2D^2 = 200 \]
Simplifying:
\[ 192 + 2D^2 = 200 \]
Therefore:
\[ 2D^2 = 200 - 192 \]
\[ 2D^2 = 8 \]
Dividing by 2:
\[ D^2 = 4 \]
Taking the square root:
\[ D = 2 \quad \text{(since D must be positive)} \]

5. **Find the Numbers**:
Now we can find the three numbers:
- First number: \( A - D = 8 - 2 = 6 \)
- Second number: \( A = 8 \)
- Third number: \( A + D = 8 + 2 = 10 \)

Thus, the three numbers are **6, 8, and 10**.

### Verification:
- **Sum**: \( 6 + 8 + 10 = 24 \) (Correct)
- **Sum of Squares**: \( 6^2 + 8^2 + 10^2 = 36 + 64 + 100 = 200 \) (Correct)

### Final Answer:
The three numbers are **6, 8, and 10**.