Sequences & Series
Sum of Series
Grade 11
Question:
<p>Let \(S(k) = 1 + 3 + 5 + \ldots + (2k-1) = 3 + k^2\). Which of the following is true?</p>
<p>(a) Principle of mathematical induction can be used to prove the formula</p>
<p>(b) <i>S</i>(<i>k</i>) < <i>S</i>(<i>k</i>+1)</p>
<p>(c) <i>S</i>(<i>k</i>) > <i>S</i>(<i>k</i>+1)</p>
<p>(d) <i>S</i>(1) is correct</p>
Step-by-Step Solution
Key Concept: Verify the formula by checking initial values before applying mathematical induction.
<p><strong>Step 1:</strong> We have $S(k) = 1 + 3 + 5 + \ldots + (2k-1) = 3 + k^2$.</p><p><strong>Step 2:</strong> Check <i>S</i>(1): <i>S</i>(1) = 1, but 3 + 1² = 4. So 1 ≠ 4, which is not true.</p><p><strong>Step 3:</strong> Check <i>S</i>(2): <i>S</i>(2) = 1 + 3 = 4, but 3 + 2² = 7. So 4 ≠ 7, which is not true.</p><p><strong>Step 4:</strong> Since the base cases fail, induction cannot be applied and the formula is incorrect. Hence, $S(k) \neq S(k+1)$ in general.</p><p>The correct statement is that <i>S</i>(<i>k</i>) > <i>S</i>(<i>k</i>+1) is false. The actual relationship shows the formula is incorrect.</p><p>∴ Answer is (c).</p>
Correct Answer: c