Question

If the sum of the coefficients of even powers of \( x \) in the expansion of \( (1 - x + x^2)^{2n} \) is 3281, then \( n \) is:

• A. 4
• B. 5
• C. 6
• D. 3

Hint:

For a polynomial \( p(x) \), the sum of coefficients of even powers in \( p(x)^n \) is \( \frac{p(1)^n + p(-1)^n}{2} \).

Answer 1

The Correct Option is A

Step 1: Formula for sum of coefficients of even powers
For a polynomial \( p(x) = 1 - x + x^2 \), the sum of coefficients of even powers in \( p(x)^m \) is: \[ S_{\text{even}} = \frac{p(1)^m + p(-1)^m}{2}. \] \[ p(1) = 1 - 1 + 1 = 1, \quad p(-1) = 1 - (-1) + 1 = 3. \] For the exponent \( m = 2n \): \[ p(1)^{2n} = 1^{2n} = 1, \quad p(-1)^{2n} = 3^{2n}, \] \[ S_{\text{even}} = \frac{1 + 3^{2n}}{2}. \]
Step 2: Solve for \( n \)
Given the sum is 3281: \[ \frac{1 + 3^{2n}}{2} = 3281 \implies 1 + 3^{2n} = 6562 \implies 3^{2n} = 6561, \] \[ 3^{2n} = 6561 = 3^8 \implies 2n = 8 \implies n = 4. \] \[ \Rightarrow n = 4. \]