In the expansion of `(1-2x+3x^2 -4x^3+…."to")^4` coefficient of `x^2` is

Options

A. 72
B. 36
C. `-36`
D. `-72`

Answer

To find the coefficient of \( x^2 \) in the expansion of \( (1 - 2x + 3x^2 - 4x^3 + \ldots)^4 \), we can follow these steps:

### Step 1: Identify the series
The series can be expressed as:
\[ f(x) = 1 - 2x + 3x^2 - 4x^3 + \ldots \]
This is an infinite series. We can rewrite it in a more manageable form.

### Step 2: Recognize the pattern
The series can be expressed as:
\[ f(x) = \sum_{n=0}^{\infty} (-1)^n (n+1) x^n \]
This series can be related to the derivative of a geometric series.

### Step 3: Use the formula for the series
The series can be derived from the function:
\[ f(x) = \frac{1}{(1+x)^2} \]
This is because the derivative of \( \frac{1}{1+x} \) gives us the series we need.

### Step 4: Raise the function to the power of 4
Now, we need to raise this function to the power of 4:
\[ f(x)^4 = \left( \frac{1}{(1+x)^2} \right)^4 = \frac{1}{(1+x)^8} \]

### Step 5: Find the coefficient of \( x^2 \)
To find the coefficient of \( x^2 \) in \( \frac{1}{(1+x)^8} \), we can use the binomial theorem:
\[ \frac{1}{(1+x)^8} = \sum_{k=0}^{\infty} \binom{-8}{k} x^k \]
The coefficient of \( x^2 \) corresponds to \( k = 2 \):
\[ \text{Coefficient of } x^2 = \binom{-8}{2} = \frac{-8 \cdot (-9)}{2!} = \frac{72}{2} = 36 \]

### Conclusion
Thus, the coefficient of \( x^2 \) in the expansion of \( (1 - 2x + 3x^2 - 4x^3 + \ldots)^4 \) is:
\[ \boxed{36} \]