Sets, Relations & Functions
Principle of Mathematical Induction
Grade 11
Question:
<p>Consider the statement: "<em>P</em>(<em>n</em>) : <em>n</em><sup>2</sup> − <em>n</em> + 41 is prime". Then which one of the following is true?</p>
<p>Both <em>P</em>(3) and <em>P</em>(5) are true.</p>
<p><em>P</em>(3) is false but <em>P</em>(5) is true.</p>
<p>Both <em>P</em>(3) and <em>P</em>(5) are false.</p>
<p><em>P</em>(5) is false but <em>P</em>(3) is true.</p>
Step-by-Step Solution
Key Concept: Test P(n) for small values and find a counterexample where n² − n + 41 is composite. The statement fails at n = 41 since 41² − 41 + 41 = 41(41) which is composite, disproving the universal claim.
<p><strong>Step 1:</strong> Test P(n) for small values:</p><p>P(1) = 1 − 1 + 41 = 41 (prime) ✓</p><p>P(2) = 4 − 2 + 41 = 43 (prime) ✓</p><p>P(3) = 9 − 3 + 41 = 47 (prime) ✓</p><p>P(4) = 16 − 4 + 41 = 53 (prime) ✓</p><p><strong>Step 2:</strong> Test critical value n = 41:</p><p>P(41) = 41² − 41 + 41 = 41² − 0 = 1681 = 41 × 41</p><p>This is composite (not prime).</p><p><strong>Step 3:</strong> Conclusion - The statement P(n) is NOT true for all natural numbers n. There exists at least one counterexample (n = 41), so the universal claim is false.</p><p>∴ Answer: A (The statement is false; it fails at n = 41)</p>
Correct Answer: A