Binomial Theorem
Sum of Coefficients
Grade 11
Question:
<p>Consider <i>(1 + x + x<sup>2</sup>)<sup>n</sup> = ∑<i>a<sub>r</sub>x<sup>r</sup></i></i>, where <i>a<sub>0</sub>, a<sub>1</sub>, a<sub>2</sub>, ..., a<sub>2n</sub></i> are real numbers and <i>n</i> is a positive integer. The value of ∑<sub><i>r</i>=0</sub><sup><i>n</i></sup> <i>ra<sub>r</sub></i> is</p>
<p>(a) \(\frac{3^n - a_n}{2}\)</p>
<p>(b) \(\frac{3^n - a_n}{2}\)</p>
<p>(c) \(\frac{a_n + 3^n}{2}\)</p>
<p>(d) \(\frac{3^n + a_n}{2}\)</p>
Step-by-Step Solution
Key Concept: Differentiate the binomial expansion and substitute specific values of <i>x</i> to find linear combinations of coefficients.
<p><strong>Solution:</strong></p><p>Differentiate <i>(1 + x + x<sup>2</sup>)<sup>n</sup> = ∑ a<sub>r</sub>x<sup>r</sup></i> with respect to <i>x</i>:</p><p><i>n(1 + x + x<sup>2</sup>)<sup>n-1</sup>(1 + 2x) = ∑ r·a<sub>r</sub>x<sup>r-1</sup></i></p><p>Set <i>x = 1</i> and solve for the desired sum.</p>
Correct Answer: a