Binomial Theorem
Coefficient Selection
Grade 11
Question:
<p>If <i>n</i> is odd, the value of ∑<sub><i>r</i>=1</sub><sup><i>(n+1)/2</i></sup> <i>a<sub>2r-1</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: Apply cube roots of unity substitution to isolate odd-indexed coefficient sums.
<p><strong>Solution:</strong></p><p>Use substitution of roots of unity in <i>(1 + x + x<sup>2</sup>)<sup>n</sup></i> to extract the sum of odd-indexed coefficients for the first half.</p>
Correct Answer: b