Find three numbers in A.P. whose sum is 12 and product is 60.

Explanation

To find three numbers in Arithmetic Progression (A.P.) whose sum is 12 and product is 60, we can follow these steps:

### Step 1: Define the three numbers in A.P.
Let the three numbers be:
- \( A - D \)
- \( A \)
- \( A + D \)

Where \( A \) is the middle term and \( D \) is the common difference.

### Step 2: Set up the equation for the sum
The sum of the three numbers is given as 12:
\[ (A - D) + A + (A + D) = 12 \]
This simplifies to:
\[ 3A = 12 \]

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

### Step 4: Set up the equation for the product
The product of the three numbers is given as 60:
\[ (A - D) \cdot A \cdot (A + D) = 60 \]
Substituting \( A = 4 \):
\[ (4 - D) \cdot 4 \cdot (4 + D) = 60 \]

### Step 5: Expand the product
Expanding the left side:
\[ 4 \cdot [(4 - D)(4 + D)] = 60 \]
Using the difference of squares:
\[ 4 \cdot (16 - D^2) = 60 \]
This simplifies to:
\[ 64 - 4D^2 = 60 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ -4D^2 = 60 - 64 \]
\[ -4D^2 = -4 \]
Dividing by -4:
\[ D^2 = 1 \]

### Step 7: Solve for \( D \)
Taking the square root of both sides:
\[ D = 1 \quad \text{or} \quad D = -1 \]

### Step 8: Find the three numbers
Now we can find the three numbers for both values of \( D \):
1. For \( D = 1 \):
- \( A - D = 4 - 1 = 3 \)
- \( A = 4 \)
- \( A + D = 4 + 1 = 5 \)
- The three numbers are \( 3, 4, 5 \).

2. For \( D = -1 \):
- \( A - D = 4 - (-1) = 5 \)
- \( A = 4 \)
- \( A + D = 4 + (-1) = 3 \)
- The three numbers are \( 5, 4, 3 \).

### Conclusion
The three numbers in A.P. are \( 3, 4, 5 \) and \( 5, 4, 3 \).

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