<p>Find the sum: \({}^4C_1 + {}^5C_2 \cdot \dfrac{1}{2} + {}^6C_3\left(\dfrac{1}{2}\right)^2 + \cdots\) to \(\infty\).</p>
Step-by-Step Solution
Key Concept: Recognize this as a sum of the form ∑(^(n+3)C_n)·(1/2)^(n-1) and use the binomial series expansion of (1+x)^m with the integral or derivative relationship to match coefficients.
<p><strong>Step 1:</strong> Identify the general term. The r-th term is ^(r+3)C_r·(1/2)^(r-1) for r = 1, 2, 3, ...</p><p><strong>Step 2:</strong> Rewrite the sum as: S = 2∑_{r=1}^{∞} ^(r+3)C_r·(1/2)^r</p><p><strong>Step 3:</strong> Use the identity: ∑_{r=0}^{∞} ^(r+3)C_r·x^r = 1/(1-x)^4 (from (1-x)^(-4) binomial expansion)</p><p><strong>Step 4:</strong> Therefore: ∑_{r=1}^{∞} ^(r+3)C_r·(1/2)^r = 1/(1/2)^4 - ^3C_0 = 16 - 1 = 15</p><p><strong>Step 5:</strong> Thus S = 2 × 15 = 30</p><p>∴ <strong>Answer: 30</strong></p>
Correct Answer: 30