The sum of three numbers in A.P. is 27, and their product is 504, find them.

Answer

To solve the problem of finding three numbers in arithmetic progression (A.P.) whose sum is 27 and product is 504, 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 Sum Equation
According to the problem, the sum of these three numbers is given as 27. Therefore, we can write the equation:
\[ (a - d) + a + (a + d) = 27 \]
This simplifies to:
\[ 3a = 27 \]

### Step 3: Solve for \( a \)
Now, we can solve for \( a \):
\[ a = \frac{27}{3} = 9 \]

### Step 4: Set Up the Product Equation
Next, we know that the product of these three numbers is 504. We can write this as:
\[ (a - d) \cdot a \cdot (a + d) = 504 \]
Using the identity \( (a - d)(a + d) = a^2 - d^2 \), we can rewrite the product as:
\[ (a^2 - d^2) \cdot a = 504 \]

### Step 5: Substitute \( a \) into the Product Equation
Now, substituting \( a = 9 \) into the product equation gives us:
\[ (9^2 - d^2) \cdot 9 = 504 \]
Calculating \( 9^2 \):
\[ (81 - d^2) \cdot 9 = 504 \]

### Step 6: Simplify the Equation
Dividing both sides by 9:
\[ 81 - d^2 = \frac{504}{9} = 56 \]

### Step 7: Solve for \( d^2 \)
Rearranging the equation gives:
\[ d^2 = 81 - 56 = 25 \]

### Step 8: Solve for \( d \)
Taking the square root of both sides:
\[ d = \pm 5 \]

### Step 9: Find the Three Numbers
Now we can find the three numbers using both values of \( d \):
1. If \( d = 5 \):
- First number: \( 9 - 5 = 4 \)
- Second number: \( 9 \)
- Third number: \( 9 + 5 = 14 \)

2. If \( d = -5 \):
- First number: \( 9 - (-5) = 14 \)
- Second number: \( 9 \)
- Third number: \( 9 + (-5) = 4 \)

In both cases, the three numbers are \( 4, 9, \) and \( 14 \).

### Final Answer
The three numbers in A.P. are \( 4, 9, \) and \( 14 \).

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