Find three numbers in A.P. whose sum is 12 and product is 48.

Explanation

To find three numbers in Arithmetic Progression (A.P.) whose sum is 12 and product is 48, we can follow these steps:

### Step 1: Define the numbers
Let the three numbers in A.P. be represented as:
- First number: \( A - D \)
- Second number: \( A \)
- Third number: \( A + D \)

### Step 2: Set up the equation for the sum
According to the problem, the sum of these three numbers is 12:
\[ (A - D) + A + (A + D) = 12 \]
Simplifying this, we have:
\[ 3A = 12 \]

### Step 3: Solve for \( A \)
Dividing both sides by 3:
\[ A = 4 \]

### Step 4: Set up the equation for the product
Next, we know the product of these numbers is 48:
\[ (A - D) \cdot A \cdot (A + D) = 48 \]
Using the identity \( (A - D)(A + D) = A^2 - D^2 \), we can rewrite the product as:
\[ (A^2 - D^2) \cdot A = 48 \]

### Step 5: Substitute \( A \) into the product equation
Substituting \( A = 4 \):
\[ (4^2 - D^2) \cdot 4 = 48 \]
Calculating \( 4^2 \):
\[ (16 - D^2) \cdot 4 = 48 \]

### Step 6: Simplify the equation
Dividing both sides by 4:
\[ 16 - D^2 = 12 \]

### Step 7: Solve for \( D^2 \)
Rearranging gives:
\[ D^2 = 16 - 12 = 4 \]

### Step 8: Solve for \( D \)
Taking the square root of both sides:
\[ D = \pm 2 \]

### Step 9: Find the three numbers
Now we can find the three numbers for both cases of \( D \):

1. **When \( D = 2 \)**:
- First number: \( A - D = 4 - 2 = 2 \)
- Second number: \( A = 4 \)
- Third number: \( A + D = 4 + 2 = 6 \)

So, the numbers are \( 2, 4, 6 \).

2. **When \( D = -2 \)**:
- First number: \( A - D = 4 - (-2) = 4 + 2 = 6 \)
- Second number: \( A = 4 \)
- Third number: \( A + D = 4 + (-2) = 4 - 2 = 2 \)

So, the numbers are \( 6, 4, 2 \).

### Conclusion
The three numbers in A.P. whose sum is 12 and product is 48 are \( 2, 4, 6 \) (or \( 6, 4, 2 \), which is the same set of numbers).

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