<p>Find the sum of the series \(1 + \dfrac{3}{2} + \dfrac{7}{4} + \dfrac{15}{8} + \dfrac{31}{16} + \cdots\) up to 20 terms.</p>
<p>\(38 + \dfrac{1}{2^{18}}\)</p>
<p>\(38 + \dfrac{1}{2^{19}}\)</p>
<p>\(40 - \dfrac{1}{2^{19}}\)</p>
<p>\(40 - \dfrac{1}{2^{20}}\)</p>
Step-by-Step Solution
Key Concept: Recognize that numerators follow the pattern 2^n - 1, allowing you to split the series into two separate summations: Σ(2^n - 1)/2^(n-1) = Σ2 - Σ2^(1-n), which simplifies using geometric series formulas.
<p><strong>Step 1:</strong> Identify the pattern in numerators: 1 = 2¹-1, 3 = 2²-1, 7 = 2³-1, 15 = 2⁴-1, 31 = 2⁵-1</p><p>The nth term is: a<sub>n</sub> = (2ⁿ - 1)/2^(n-1)</p><p><strong>Step 2:</strong> Decompose each term:</p><p>a<sub>n</sub> = (2ⁿ - 1)/2^(n-1) = 2ⁿ/2^(n-1) - 1/2^(n-1) = 2 - (1/2)^(n-1)</p><p><strong>Step 3:</strong> Sum the first 20 terms:</p><p>S₂₀ = Σ(n=1 to 20)[2 - (1/2)^(n-1)]</p><p>= 2(20) - Σ(n=1 to 20)(1/2)^(n-1)</p><p><strong>Step 4:</strong> The second sum is a geometric series with a = 1, r = 1/2, and 20 terms:</p><p>Σ(n=1 to 20)(1/2)^(n-1) = (1 - (1/2)²⁰)/(1 - 1/2) = 2(1 - (1/2)²⁰) = 2 - 2^(-19)</p><p><strong>Step 5:</strong> Therefore:</p><p>S₂₀ = 40 - (2 - 2^(-19)) = 38 + 2^(-19) = 38 + 1/2^19</p><p>∴ Answer: B</p>
Correct Answer: B