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><em>S</em>(1) is correct.</p>
<p>Principle of mathematical induction can be used to prove the formula.</p>
<p>\(S(K) \neq S(K+1)\).</p>
<p>\(S(K) = S(K+1)\).</p>
Step-by-Step Solution
Key Concept: Recognize that S(K) represents the sum of first K odd numbers, which has a well-known formula K². The given equation S(K) = 3 + K² is false for most values, so we must find which statement about this inconsistency is true.
<p><strong>Step 1:</strong> Recall that the sum of first K odd numbers is: 1 + 3 + 5 + ⋯ + (2K−1) = K²</p><p><strong>Step 2:</strong> The given statement claims S(K) = 3 + K². This means K² = 3 + K², which simplifies to 0 = 3 (a contradiction).</p><p><strong>Step 3:</strong> This equation has NO solution for any positive integer K. The statement S(K) = 3 + K² is false for all K ∈ ℕ.</p><p><strong>Step 4:</strong> The true statement must be one that correctly identifies this impossibility or states the correct formula. The correct answer identifies that the given equation cannot be satisfied.</p><p>∴ Answer: D</p>
Correct Answer: D