<p>Given \(S = 2\,{}^{20}C_0 + 5\,{}^{20}C_1 + 8\,{}^{20}C_2 + \cdots + 62\,{}^{20}C_{20}\), find the value of \(S\).</p>
Step-by-Step Solution
Key Concept: Recognize that the coefficients 2, 5, 8, ..., 62 form an arithmetic progression with first term 2 and common difference 3, allowing you to express the general term as (3k+2)·C(20,k) and decompose S into two separate binomial sums.
<p><strong>Step 1:</strong> Identify the pattern in coefficients. The terms are 2, 5, 8, ..., 62, which form an AP with first term a = 2 and common difference d = 3. The general term is 3k + 2 (for k = 0, 1, 2, ..., 20).</p><p><strong>Step 2:</strong> Write S = Σ(k=0 to 20) (3k+2)·C(20,k) = 3·Σ(k=0 to 20) k·C(20,k) + 2·Σ(k=0 to 20) C(20,k)</p><p><strong>Step 3:</strong> Use binomial identities:</p><p>• Σ(k=0 to 20) C(20,k) = 2^20 (standard binomial expansion)</p><p>• For Σ(k=0 to 20) k·C(20,k): Use the identity k·C(n,k) = n·C(n-1,k-1). Thus Σ(k=0 to 20) k·C(20,k) = 20·Σ(j=0 to 19) C(19,j) = 20·2^19</p><p><strong>Step 4:</strong> Substitute back: S = 3(20·2^19) + 2(2^20) = 60·2^19 + 2·2^20 = 60·2^19 + 4·2^19 = 64·2^19 = 2^6·2^19 = 2^25</p><p>∴ Answer: <strong>2^25</strong></p>
Correct Answer: 2^25