Sets, Relations & Functions
Principle of Mathematical Induction
Grade 11
Question:
<p>Let <em>S</em>(<em>K</em>) = 1 + 3 + 5 + ⋯ + (2<em>K</em> − 1) = 3 + <em>K</em><sup>2</sup>. Then which of the following is true?</p><p>(1) <em>S</em>(1) is not true<br>(2) Principle of Mathematical Induction cannot be used<br>(3) <em>S</em>(<em>K</em>) ≠ <em>S</em>(<em>K</em> + 1)<br>(4) <em>S</em>(<em>K</em>) ⟹ <em>S</em>(<em>K</em> + 1)</p>
<p>(1) <em>S</em>(1) is not true and Principle of Mathematical Induction can be used</p>
<p>(2) Principle of Mathematical Induction cannot be used since <em>S</em>(1) is not true</p>
<p>(3) <em>S</em>(<em>K</em>) ≠ <em>S</em>(<em>K</em> + 1)</p>
<p>(4) <em>S</em>(<em>K</em>) ⟹ <em>S</em>(<em>K</em> + 1)</p>
Step-by-Step Solution
Key Concept: Verify the base case S(1) and check if the inductive step S(K) ⟹ S(K+1) holds logically, even though the formula itself is incorrect. The statement tests understanding that PMI works on logical implication structure, not on correctness of the final formula.
<p><strong>Step 1: Check Base Case S(1)</strong></p><p>S(1) = 1 (sum of first odd number) should equal 1² = 1</p><p>Given formula gives: S(1) = 3 + 1² = 4 ✗</p><p>So S(1) is FALSE, making option (1) TRUE. But this is not the answer.</p><p><strong>Step 2: Analyze Option (4) - S(K) ⟹ S(K+1)</strong></p><p>Assume S(K) is true: 1 + 3 + 5 + ⋯ + (2K−1) = 3 + K²</p><p>Then: S(K+1) = 1 + 3 + 5 + ⋯ + (2K−1) + (2K+1)</p><p>= [3 + K²] + (2K+1) = K² + 2K + 4 = (K+1)² + 3 ✓</p><p>This matches the formula structure for S(K+1).</p><p><strong>Step 3: Logical Conclusion</strong></p><p>The conditional statement "IF S(K) THEN S(K+1)" is logically TRUE (the implication holds), even though the base case fails. Option (2) is false because PMI structure can be applied; option (3) is false because S(K) ⟹ S(K+1) does hold.</p><p>∴ Answer: (4) S(K) ⟹ S(K + 1) is true</p>
Correct Answer: C