Binomial Theorem
Properties of Binomial Coefficients
Grade 11
Question:
<p>If <span>\(\sum_{r=0}^{n} {}^n C_r a^r b^{n-r} = na(a+b)^{n-1}\)</span>, then find the value of the expression. We have,<br>\[\sum_{r=0}^{n} r \cdot {}^n C_r a^r b^{n-r}\]</p>
<p>\(na(a+b)^{n-1}\)</p>
<p>\(na(a+b)^n\)</p>
<p>\(n(a+b)^{n-1}\)</p>
<p>\(na^{n-1}(a+b)\)</p>
Step-by-Step Solution
Key Concept: Recognize that ∑r·ⁿCᵣaʳbⁿ⁻ʳ is the derivative of the binomial expansion (a+b)ⁿ with respect to one variable, which can be obtained by differentiating aʳ in the given identity.
<p><strong>Step 1:</strong> Start with the binomial expansion identity: (a+b)ⁿ = ∑ᵣ₌₀ⁿ ⁿCᵣaʳbⁿ⁻ʳ</p><p><strong>Step 2:</strong> Differentiate both sides with respect to 'a': d/da[(a+b)ⁿ] = d/da[∑ᵣ₌₀ⁿ ⁿCᵣaʳbⁿ⁻ʳ]</p><p><strong>Step 3:</strong> Left side: n(a+b)ⁿ⁻¹</p><p><strong>Step 4:</strong> Right side: ∑ᵣ₌₀ⁿ ⁿCᵣ·r·aʳ⁻¹·bⁿ⁻ʳ = (1/a)∑ᵣ₌₀ⁿ r·ⁿCᵣaʳbⁿ⁻ʳ</p><p><strong>Step 5:</strong> Therefore: n(a+b)ⁿ⁻¹ = (1/a)∑ᵣ₌₀ⁿ r·ⁿCᵣaʳbⁿ⁻ʳ</p><p><strong>Step 6:</strong> Multiply both sides by 'a': ∑ᵣ₌₀ⁿ r·ⁿCᵣaʳbⁿ⁻ʳ = <strong>na(a+b)ⁿ⁻¹</strong></p><p>∴ Answer: A</p>
Correct Answer: A