<p>The sum of the first 20 terms of the series \(1 + \dfrac{3}{2} + \dfrac{7}{4} + \dfrac{15}{8} + \dfrac{31}{16} + \cdots\), is</p>
<p>\(38 + \dfrac{2}{2^{19}}\)</p>
<p>\(38 + \dfrac{2}{2^{20}}\)</p>
<p>\(39 + \dfrac{1}{2^{20}}\)</p>
<p>\(39 + \dfrac{1}{2^{19}}\)</p>
Step-by-Step Solution
Key Concept: Recognize that each numerator follows the pattern 2^n - 1, making the general term a_n = (2^n - 1)/2^(n-1). Split into two separate series: Σ(2^n/2^(n-1)) - Σ(1/2^(n-1)), which simplifies to 2Σ2 - Σ(1/2)^(n-1) (geometric series).
<p><strong>Step 1: Identify the general term</strong></p><p>Numerators: 1, 3, 7, 15, 31, ... = 2¹-1, 2²-1, 2³-1, 2⁴-1, 2⁵-1, ...</p><p>Denominators: 1, 2, 4, 8, 16, ... = 2⁰, 2¹, 2², 2³, 2⁴, ...</p><p>General term: a<sub>n</sub> = (2ⁿ - 1)/2^(n-1)</p><p><strong>Step 2: Decompose the series</strong></p><p>S₂₀ = Σ(n=1 to 20) [(2ⁿ - 1)/2^(n-1)]</p><p>= Σ(n=1 to 20) [2ⁿ/2^(n-1)] - Σ(n=1 to 20) [1/2^(n-1)]</p><p>= Σ(n=1 to 20) 2 - Σ(n=1 to 20) (1/2)^(n-1)</p><p><strong>Step 3: Evaluate each sum</strong></p><p>First sum: Σ(n=1 to 20) 2 = 2 × 20 = 40</p><p>Second sum: Geometric series with a = 1, r = 1/2, n = 20</p><p>= [1 - (1/2)²⁰]/[1 - 1/2] = 2[1 - 1/2²⁰] = 2 - 1/2¹⁹</p><p><strong>Step 4: Calculate final answer</strong></p><p>S₂₀ = 40 - (2 - 1/2¹⁹) = 38 + 1/2¹⁹</p><p>∴ Answer: C</p>
Correct Answer: C