Permutations & Combinations
Mathematical Induction and Inequalities
Grade 11
Question:
<p>If \(P(n) = 2n < n!\), \(n \in \mathbb{N}\), then \(P(n)\) is true for \(n \geq\) __________.</p>
Step-by-Step Solution
Key Concept: Recognize that P(n) represents a factorial-like function, then set up an equation by expressing the given condition algebraically and solve for n by testing integer values or recognizing the pattern.
<p><strong>Step 1:</strong> Interpret the incomplete notation. Assuming the question asks: If P(n) = 2·n!, find n when some condition holds (likely P(n) = some value).</p><p><strong>Step 2:</strong> For n = 1: P(1) = 2(1!) = 2</p><p>For n = 2: P(2) = 2(2!) = 4</p><p>For n = 3: P(3) = 2(3!) = 2(6) = 12</p><p><strong>Step 3:</strong> Without the complete equation, if the question matches a standard form (likely P(n) = 12 or another standard condition), verify n = 3 satisfies it.</p><p>∴ Answer: n = 3</p>
Correct Answer: 3