Sum of coefficients of terms of even powers of x in `(1 - x + x^2 - x^3)^7` is

Options

A. `-2^17`
B. `2^17`
C. `2^13`
D. `-2^13`

Answer

To find the sum of the coefficients of the terms of even powers of \( x \) in the expression \( (1 - x + x^2 - x^3)^7 \), we can follow these steps:

### Step 1: Define the function
Let
\[ f(x) = (1 - x + x^2 - x^3)^7 \]

### Step 2: Use the property of even functions
The sum of the coefficients of the even powers of \( x \) in a polynomial can be calculated using the formula:
\[ \text{Sum of coefficients of even powers} = \frac{f(1) + f(-1)}{2} \]

### Step 3: Calculate \( f(1) \)
Now, we calculate \( f(1) \):
\[ f(1) = (1 - 1 + 1^2 - 1^3)^7 = (1 - 1 + 1 - 1)^7 = (0)^7 = 0 \]

### Step 4: Calculate \( f(-1) \)
Next, we calculate \( f(-1) \):
\[ f(-1) = (1 - (-1) + (-1)^2 - (-1)^3)^7 = (1 + 1 + 1 + 1)^7 = (4)^7 = 2^{14} \]

### Step 5: Apply the formula
Now, we can substitute \( f(1) \) and \( f(-1) \) into the formula:
\[ \text{Sum of coefficients of even powers} = \frac{f(1) + f(-1)}{2} = \frac{0 + 2^{14}}{2} = \frac{2^{14}}{2} = 2^{13} \]

### Final Answer
Thus, the sum of the coefficients of the terms of even powers of \( x \) in the expression \( (1 - x + x^2 - x^3)^7 \) is
\[ \boxed{2^{13}} \]

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