The sum of three numbers in AP is 12 and sum of their cubes is 408 find the product of the numbers:

Options

A. 17
B. 35
C. 28
D. 36

Answer

To solve the problem step by step, we can follow these instructions:

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

### Step 2: Set up the equation for the sum of the numbers
According to the problem, the sum of these three numbers is 12:
\[ (a - d) + a + (a + d) = 12 \]
This simplifies to:
\[ 3a = 12 \]
Thus, we can solve for \( a \):
\[ a = \frac{12}{3} = 4 \]

### Step 3: Substitute \( a \) back into the expressions for the numbers
Now we substitute \( a = 4 \):
- First number: \( 4 - d \)
- Second number: \( 4 \)
- Third number: \( 4 + d \)

### Step 4: Set up the equation for the sum of the cubes
The problem states that the sum of their cubes is 408:
\[ (4 - d)^3 + 4^3 + (4 + d)^3 = 408 \]
Calculating \( 4^3 \):
\[ 4^3 = 64 \]
So we have:
\[ (4 - d)^3 + 64 + (4 + d)^3 = 408 \]
This simplifies to:
\[ (4 - d)^3 + (4 + d)^3 = 408 - 64 = 344 \]

### Step 5: Use the identity for the sum of cubes
Using the identity \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \), where \( x = 4 - d \) and \( y = 4 + d \):
\[ (4 - d + 4 + d)((4 - d)^2 - (4 - d)(4 + d) + (4 + d)^2) = 344 \]
This simplifies to:
\[ 8((4 - d)^2 - (4^2 - d^2) + (4 + d)^2) = 344 \]
Calculating \( (4 - d)^2 + (4 + d)^2 \):
\[ (4 - d)^2 = 16 - 8d + d^2 \]
\[ (4 + d)^2 = 16 + 8d + d^2 \]
Adding these:
\[ (4 - d)^2 + (4 + d)^2 = 32 + 2d^2 \]
Now substituting back:
\[ 8((32 + 2d^2) - (16 - d^2)) = 344 \]
This simplifies to:
\[ 8(16 + 3d^2) = 344 \]
Dividing both sides by 8:
\[ 16 + 3d^2 = 43 \]
Thus:
\[ 3d^2 = 27 \quad \Rightarrow \quad d^2 = 9 \]
So:
\[ d = 3 \quad \text{or} \quad d = -3 \]

### Step 6: Find the three numbers for both values of \( d \)
1. If \( d = 3 \):
- First number: \( 4 - 3 = 1 \)
- Second number: \( 4 \)
- Third number: \( 4 + 3 = 7 \)

2. If \( d = -3 \):
- First number: \( 4 - (-3) = 7 \)
- Second number: \( 4 \)
- Third number: \( 4 + (-3) = 1 \)

### Step 7: Calculate the product of the numbers
In both cases, the numbers are \( 1, 4, 7 \):
\[ \text{Product} = 1 \times 4 \times 7 = 28 \]

### Final Answer
The product of the three numbers is \( 28 \).

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