Coefficient of `x^49` in the product (x - 99) (x - 97) _____ (x - 3) (x - 1) is

Options

A. -4900
B. -2400
C. -2500
D. `-98^2`

Answer

To find the coefficient of \( x^{49} \) in the product \( (x - 99)(x - 97) \cdots (x - 3)(x - 1) \), we can follow these steps:

### Step 1: Identify the terms in the product
The product consists of terms from \( x - 1 \) to \( x - 99 \) with a common difference of 2. The terms are:
\[ x - 1, x - 3, x - 5, \ldots, x - 99 \]

### Step 2: Determine the number of terms
This sequence is an arithmetic progression (AP) where:
- First term \( a = 1 \)
- Last term \( l = 99 \)
- Common difference \( d = 2 \)

To find the number of terms \( n \):
\[ l = a + (n - 1)d \]
Substituting the values:
\[ 99 = 1 + (n - 1) \cdot 2 \]
\[ 99 - 1 = (n - 1) \cdot 2 \]
\[ 98 = (n - 1) \cdot 2 \]
\[ n - 1 = \frac{98}{2} = 49 \]
\[ n = 50 \]

### Step 3: Coefficient of \( x^{49} \)
The coefficient of \( x^{49} \) in the expansion of the product can be found by taking the constant terms from one of the factors. Since there are 50 terms, the coefficient of \( x^{49} \) will be the negative sum of the constant terms taken one at a time.

### Step 4: Calculate the sum of the constant terms
The constant terms are \( -1, -3, -5, \ldots, -99 \). The sum of these terms can be calculated as:
\[ \text{Sum} = - (1 + 3 + 5 + \ldots + 99) \]
The series \( 1 + 3 + 5 + \ldots + 99 \) is also an arithmetic series where:
- First term \( a = 1 \)
- Last term \( l = 99 \)
- Number of terms \( n = 50 \)

The sum of the first \( n \) odd numbers is given by:
\[ \text{Sum} = n^2 \]
Thus,
\[ \text{Sum} = 50^2 = 2500 \]

### Step 5: Final coefficient
Therefore, the coefficient of \( x^{49} \) is:
\[ - (1 + 3 + 5 + \ldots + 99) = -2500 \]

### Conclusion
The coefficient of \( x^{49} \) in the product \( (x - 99)(x - 97) \cdots (x - 3)(x - 1) \) is:
\[ \boxed{-2500} \]