If sum of 5 consecutive terms of an A.P. is 25 & product of these terms is 2520. If one of the terms is -1/2 then the value of greatest term is.

Options

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

Answer

To solve the problem step by step, we will follow the given information about the arithmetic progression (A.P.) and derive the necessary values.

### Step 1: Define the Terms of the A.P.
Let the five consecutive terms of the A.P. be:
- \( a - 2d \)
- \( a - d \)
- \( a \)
- \( a + d \)
- \( a + 2d \)

### Step 2: Use the Sum of the Terms
According to the problem, the sum of these five terms is given as 25:
\[ (a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 25 \]
Simplifying this:
\[ 5a = 25 \]
Thus, we find:
\[ a = 5 \]

### Step 3: Use the Product of the Terms
Next, we know the product of these terms is 2520:
\[ (a - 2d)(a - d)(a)(a + d)(a + 2d) = 2520 \]
Substituting \( a = 5 \):
\[ (5 - 2d)(5 - d)(5)(5 + d)(5 + 2d) = 2520 \]

### Step 4: Simplify the Product
We can express the product as:
\[ (5^2 - (2d)^2)(5^2 - d^2) \cdot 5 \]
This simplifies to:
\[ (25 - 4d^2)(25 - d^2) \cdot 5 = 2520 \]
Expanding this gives:
\[ 5[(25 - 4d^2)(25 - d^2)] = 2520 \]
Dividing both sides by 5:
\[ (25 - 4d^2)(25 - d^2) = 504 \]

### Step 5: Expand and Rearrange
Expanding the left side:
\[ 625 - 25d^2 - 100d^2 + 4d^4 = 504 \]
Combining like terms:
\[ 4d^4 - 125d^2 + 121 = 0 \]

### Step 6: Substitute \( x = d^2 \)
Let \( x = d^2 \):
\[ 4x^2 - 125x + 121 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{125 \pm \sqrt{(-125)^2 - 4 \cdot 4 \cdot 121}}{2 \cdot 4} \]
Calculating the discriminant:
\[ = \frac{125 \pm \sqrt{15625 - 1936}}{8} \]
\[ = \frac{125 \pm \sqrt{13689}}{8} \]
\[ = \frac{125 \pm 117}{8} \]

Calculating the two possible values:
1. \( x = \frac{242}{8} = 30.25 \)
2. \( x = \frac{8}{8} = 1 \)

### Step 8: Find \( d \)
Since \( x = d^2 \):
1. \( d^2 = 30.25 \) gives \( d = \pm \sqrt{30.25} = \pm \frac{11}{2} \)
2. \( d^2 = 1 \) gives \( d = \pm 1 \)

### Step 9: Determine the Terms
Now substituting back to find the terms:
1. If \( d = \frac{11}{2} \):
- \( 5 - 2 \cdot \frac{11}{2} = -6 \)
- \( 5 - \frac{11}{2} = -\frac{1}{2} \)
- \( 5 \)
- \( 5 + \frac{11}{2} = \frac{21}{2} \)
- \( 5 + 2 \cdot \frac{11}{2} = 16 \)

2. If \( d = 1 \):
- \( 5 - 2 = 3 \)
- \( 5 - 1 = 4 \)
- \( 5 \)
- \( 5 + 1 = 6 \)
- \( 5 + 2 = 7 \)

### Step 10: Identify the Greatest Term
From the first case, the terms are:
- \( -6, -\frac{1}{2}, 5, \frac{21}{2}, 16 \)

The greatest term is:
\[ \text{Greatest term} = 16 \]

### Conclusion
The value of the greatest term is \( 16 \).