Binomial Theorem
Coefficient Selection
Grade 11
Question:
<p>If <i>n</i> is even, the value of ∑<sub><i>r</i>=0</sub><sup><i>n/2 - 1</i></sup> <i>a<sub>2r</sub></i> is</p>
<p>(a) \(\frac{3^n - 1 + a_n}{2}\)</p>
<p>(b) \(\frac{3^n - 1 - a_n}{4}\)</p>
<p>(c) \(\frac{3 + 1 - a_n}{2}\)</p>
<p>(d) \(\frac{3 + 1 - 2a_n}{4}\)</p>
Step-by-Step Solution
Key Concept: Substitute cube roots of unity to separate odd and even indexed coefficients in the expansion.
<p><strong>Solution:</strong></p><p>Use the expansions at <i>x = 1</i>, <i>x = -1</i>, and <i>x = ω</i> (where <i>ω</i> is a cube root of unity) to isolate the sum of even-indexed coefficients.</p>
Correct Answer: a