<p>The sum \(1 + 3 + 7 + 15 + 31 + \cdots\) to 100 terms is</p>
Step-by-Step Solution
Key Concept: Recognize that each term follows the pattern 2^n - 1, then use summation formulas to convert the sum into a telescoping or closed form involving powers of 2.
<p><strong>Step 1: Identify the pattern</strong></p><p>The terms are: 1 = 2¹-1, 3 = 2²-1, 7 = 2³-1, 15 = 2⁴-1, 31 = 2⁵-1</p><p>General term: aₙ = 2ⁿ - 1</p><p><strong>Step 2: Set up the sum</strong></p><p>S₁₀₀ = Σ(2ⁿ - 1) from n=1 to 100</p><p>= Σ2ⁿ - Σ1 from n=1 to 100</p><p><strong>Step 3: Evaluate each part</strong></p><p>Σ2ⁿ = 2¹ + 2² + 2³ + ... + 2¹⁰⁰ = 2(2¹⁰⁰ - 1)/(2-1) = 2¹⁰¹ - 2</p><p>Σ1 = 100</p><p><strong>Step 4: Combine results</strong></p><p>S₁₀₀ = (2¹⁰¹ - 2) - 100 = 2¹⁰¹ - 102</p><p>∴ Answer: C (2¹⁰¹ - 102)</p>
Correct Answer: C