Question

In the expansion of (1 - 3x + 3x² - x³)²ⁿ, the middle term is

• A. \( (n+1)^\text{th} \) term
• B. \( (2n+1)^\text{th} \) term
• C. \( (3n+1)^\text{th} \) term
• D. None of these

Hint:

Middle term of polynomial of degree 3 raised to power \( 2n \) is \( (3n + 1)^\text{th} \) term.

Answer 1

The Correct Option is C

We are given the expression: \[ (1 - 3x + 3x^2 - x^3)^{2n} \] Let: \[ f(x) = (1 - 3x + 3x^2 - x^3) \] Observe that: \[ f(x) = (1 - x)^3 \] This is because: \[ (1 - x)^3 = 1 - 3x + 3x^2 - x^3 \] So the given expression becomes: \[ [(1 - x)^3]^{2n} = (1 - x)^{6n} \] Now, the number of terms in the expansion of \( (1 - x)^{6n} \) is: \[ 6n + 1 \] Hence, the middle term of a binomial expansion with odd number of terms is: \[ \left( \frac{6n + 1}{2} \right)^\text{th} \text{ term} = (3n + 1)^\text{th} \text{ term} \]