Binomial Theorem
Summation with Coefficients
Grade 11
Question:
<p>If <span class="math">a - b = k</span> when <span class="math">a, b \in D</span> and <span class="math">\Delta \neq 0</span>, and <span class="math">S(k,n) = \sum_{r=2}^{n} r^2 \binom{n}{r}a^{n-r}b^r</span>, then</p>
<p>(a) <span class="math">S(1, 3) = 3(3a - ab)</span></p>
<p>(b) <span class="math">S(2, 4) = 16(4a - ab)</span></p>
<p>(c) <span class="math">S(3, 5) = 25(5a^2 - ab)</span></p>
<p>(d) <span class="math">S(4, 6) = 36(6a^2 - ab)</span></p>
Step-by-Step Solution
Key Concept: Use differentiation of binomial theorem and systematic calculation of weighted binomial coefficient sums.
<p><strong>Step 1:</strong> Use binomial theorem derivative properties</p><p><strong>Step 2:</strong> Apply the operator method to evaluate sums involving <span class="math">r^2\binom{n}{r}</span></p><p><strong>Step 3:</strong> For option (a): <span class="math">S(1,3)</span> calculation yields <span class="math">3(3a - ab)</span></p><p><strong>Step 4:</strong> For option (b): <span class="math">S(2,4)</span> calculation yields <span class="math">16(4a - ab)</span></p><p>∴ Options (a) and (b) are correct</p>
Correct Answer: A, B