Four different integers form an increasing A.P .One of these numbers is equal to the sum of the squares of the other three numbers. Then
The product of all numbers is

Options

A. 5
B. 10
C. 20
D. 17

Explanation

To solve the problem, we need to find four different integers that form an increasing arithmetic progression (A.P.) such that one of the numbers is equal to the sum of the squares of the other three numbers. Let's denote the four integers in the A.P. as follows:

Let the integers be:
1. \( a - d \)
2. \( a \)
3. \( a + d \)
4. \( a + 2d \)

### Step 1: Set up the equation
According to the problem, one of these numbers is equal to the sum of the squares of the other three. We can assume that \( a + 2d \) is the number equal to the sum of the squares of the other three. Thus, we can write the equation:

\[ a + 2d = (a - d)^2 + a^2 + (a + d)^2 \]

### Step 2: Expand the squares
Now, we will expand the right-hand side of the equation:

\[ (a - d)^2 = a^2 - 2ad + d^2 \]
\[ a^2 = a^2 \]
\[ (a + d)^2 = a^2 + 2ad + d^2 \]

Adding these together:

\[ (a - d)^2 + a^2 + (a + d)^2 = (a^2 - 2ad + d^2) + a^2 + (a^2 + 2ad + d^2) \]

This simplifies to:

\[ 3a^2 + 2d^2 \]

### Step 3: Set up the equation
Now, substituting back into our original equation, we have:

\[ a + 2d = 3a^2 + 2d^2 \]

### Step 4: Rearranging the equation
Rearranging gives us:

\[ 3a^2 - a + 2d^2 - 2d = 0 \]

### Step 5: Solve for \( d \)
This is a quadratic equation in terms of \( d \). We can use the quadratic formula to solve for \( d \):

\[ d = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 2 \cdot (3a^2 - a)}}{2 \cdot 2} \]

This simplifies to:

\[ d = \frac{2 \pm \sqrt{4 - 8(3a^2 - a)}}{4} \]

### Step 6: Finding integer solutions
To find integer solutions for \( a \) and \( d \), we can test various integer values for \( a \).

After testing integer values, we find that:

If \( a = 0 \), then \( d = 1 \) leads us to the integers:

1. \( a - d = -1 \)
2. \( a = 0 \)
3. \( a + d = 1 \)
4. \( a + 2d = 2 \)

### Step 7: Calculate the product
Now, we can calculate the product of these integers:

\[ (-1) \times 0 \times 1 \times 2 = 0 \]

### Conclusion
Thus, the product of all four numbers is:

\[ \boxed{0} \]