<p>The number of terms in the expansion of \((1 + 2x + x^2)^n\) when expanded in descending powers of \(x\), is</p>
Step-by-Step Solution
Key Concept: Recognize that (1 + 2x + x²)ⁿ = [(1 + x)²]ⁿ = (1 + x)²ⁿ, which has a fixed maximum power. The number of distinct terms equals the highest power of x plus 1.
<p><strong>Step 1:</strong> Recognize the structure of the expression.</p><p>(1 + 2x + x²)ⁿ = [(1 + x)²]ⁿ = (1 + x)²ⁿ</p><p><strong>Step 2:</strong> Find the expansion of (1 + x)²ⁿ using binomial theorem.</p><p>(1 + x)²ⁿ = C(2n,0) + C(2n,1)x + C(2n,2)x² + ... + C(2n,2n)x²ⁿ</p><p><strong>Step 3:</strong> Count the number of terms.</p><p>The expansion contains terms from x⁰ to x²ⁿ, giving us (2n + 1) distinct terms.</p><p>∴ Answer: D (2n + 1)</p>
Correct Answer: D