The coefficient of `x^(99)` in `(x + 1) (x + 3) (x + 5)…(x + 199)` is

Options

A. `1 + 2 + 3 + …+ 99`
B. `1 + 3 + 5 + … + 199`
C. `1.3.5..199`
D. none of these

Answer

To find the coefficient of \( x^{99} \) in the expression \( (x + 1)(x + 3)(x + 5) \ldots (x + 199) \), we can follow these steps:

### Step 1: Identify the number of terms
The expression consists of terms from \( x + 1 \) to \( x + 199 \) with a common difference of 2. This means we have the sequence of odd numbers starting from 1 to 199.

### Step 2: Count the total number of terms
The sequence of odd numbers can be represented as:
1, 3, 5, ..., 199.

To find the number of terms, we can use the formula for the \( n \)-th term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]
where \( a = 1 \), \( d = 2 \), and \( a_n = 199 \).

Setting up the equation:
\[ 199 = 1 + (n-1) \cdot 2 \]
\[ 198 = (n-1) \cdot 2 \]
\[ n-1 = 99 \implies n = 100 \]

So, there are 100 terms in total.

### Step 3: Determine the coefficient of \( x^{99} \)
To get \( x^{99} \), we need to select \( x \) from 99 of the 100 factors and the constant term from one of the factors. This means we will be choosing one constant term from one of the 100 factors.

### Step 4: Calculate the sum of the constant terms
The constant terms in the factors are:
1, 3, 5, ..., 199.

The sum of these constant terms can be calculated using the formula for the sum of the first \( n \) odd numbers:
\[ \text{Sum} = n^2 \]
where \( n \) is the number of terms. Since we have 100 terms:
\[ \text{Sum} = 100^2 = 10000 \]

### Step 5: Conclusion
Thus, the coefficient of \( x^{99} \) in the expression \( (x + 1)(x + 3)(x + 5) \ldots (x + 199) \) is \( 10000 \).

### Final Answer:
The coefficient of \( x^{99} \) is \( 10000 \).

---