Three numbers are in AP. If their sum is 27 and their product is 648, find the numbers/

Answer

To find the three numbers in Arithmetic Progression (AP) given their sum and product, we can follow these steps:

### Step 1: Define the Numbers
Let the three numbers in AP be represented as:
- First number: \( a - d \)
- Second number: \( a \)
- Third number: \( a + d \)

### Step 2: Set Up the Sum Equation
According to the problem, the sum of the three numbers is 27:
\[ (a - d) + a + (a + d) = 27 \]
Simplifying this, we get:
\[ 3a = 27 \]
From this, we can solve for \( a \):
\[ a = \frac{27}{3} = 9 \]

### Step 3: Set Up the Product Equation
The product of the three numbers is given as 648:
\[ (a - d) \cdot a \cdot (a + d) = 648 \]
Substituting \( a = 9 \):
\[ (9 - d) \cdot 9 \cdot (9 + d) = 648 \]
This simplifies to:
\[ 9 \cdot ((9 - d)(9 + d)) = 648 \]
Using the difference of squares:
\[ 9 \cdot (81 - d^2) = 648 \]
Dividing both sides by 9:
\[ 81 - d^2 = \frac{648}{9} = 72 \]
Rearranging gives:
\[ d^2 = 81 - 72 = 9 \]

### Step 4: Solve for \( d \)
Taking the square root of both sides:
\[ d = \sqrt{9} = 3 \]
Thus, \( d \) can be either \( 3 \) or \( -3 \).

### Step 5: Find the Numbers
Now we can find the three numbers:
1. If \( d = 3 \):
- First number: \( 9 - 3 = 6 \)
- Second number: \( 9 \)
- Third number: \( 9 + 3 = 12 \)

2. If \( d = -3 \):
- First number: \( 9 - (-3) = 12 \)
- Second number: \( 9 \)
- Third number: \( 9 + (-3) = 6 \)

### Conclusion
The three numbers are:
\[ 6, 9, 12 \quad \text{or} \quad 12, 9, 6 \]