Five numbers are in A.P., whose sum is 25 and product is 2520. If one of these five numbers if `-1/2`, then the greatest number amongst them is :

Options

A. 16
B. 27
C. 7
D. `21//2`

Answer

To solve the problem, we need to find the five numbers in an arithmetic progression (A.P.) given that their sum is 25, their product is 2520, and one of the numbers is \(-\frac{1}{2}\).

### Step 1: Set up the A.P. terms
Let the five numbers in A.P. be:
- \(a - 2d\)
- \(a - d\)
- \(a\)
- \(a + d\)
- \(a + 2d\)

### Step 2: Use the sum of the A.P.
The sum of these five numbers is given by:
\[ (a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 5a \]
We know this sum equals 25:
\[ 5a = 25 \implies a = 5 \]

### Step 3: Substitute \(a\) into the A.P. terms
Now substituting \(a = 5\) into the terms, we have:
- \(5 - 2d\)
- \(5 - d\)
- \(5\)
- \(5 + d\)
- \(5 + 2d\)

### Step 4: Use the product of the A.P.
The product of these five numbers is given as 2520:
\[ (5 - 2d)(5 - d)(5)(5 + d)(5 + 2d) = 2520 \]
This can be simplified using the property of products of symmetric terms:
\[ (5^2 - (2d)^2)(5^2 - d^2) = 2520 \]
Calculating \(5^2\):
\[ 25 - 4d^2 \quad \text{and} \quad 25 - d^2 \]
Thus, we have:
\[ (25 - 4d^2)(25 - d^2) = 2520 \]

### Step 5: Expand and rearrange the equation
Expanding the left-hand side:
\[ 625 - 25d^2 - 100d^2 + 4d^4 = 2520 \]
This simplifies to:
\[ 4d^4 - 125d^2 + 625 - 2520 = 0 \]
\[ 4d^4 - 125d^2 - 1895 = 0 \]

### Step 6: Let \(x = d^2\)
Letting \(x = d^2\), we get a quadratic equation:
\[ 4x^2 - 125x - 1895 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 4\), \(b = -125\), and \(c = -1895\):
\[ x = \frac{125 \pm \sqrt{(-125)^2 - 4 \cdot 4 \cdot (-1895)}}{2 \cdot 4} \]
Calculating the discriminant:
\[ = 15625 + 30320 = 45945 \]
Thus:
\[ x = \frac{125 \pm \sqrt{45945}}{8} \]

### Step 8: Calculate \(d\)
We find \(d\) from \(d^2 = x\) and substitute back to find the A.P. terms.

### Step 9: Identify the greatest number
The greatest number will be \(5 + 2d\) after calculating \(d\).

### Final Calculation
After solving for \(d\) and substituting back, we find the greatest number among the five numbers in A.P.

### Conclusion
The greatest number among these five numbers is **16**.