Basic Mathematics & Logarithm
Mathematical Induction
Grade 11
Question:
<p>The value of the natural number n such that the inequality <latex>2^n > 2n + 1</latex> is valid, is</p>
<p>(a) for <latex>n \geq 3</latex></p>
<p>(b) for <latex>n < 3</latex></p>
<p>(c) for all n</p>
<p>(d) for some n</p>
Step-by-Step Solution
Key Concept: Use mathematical induction to establish that the exponential function grows faster than linear functions beyond a certain threshold.
<p>Testing values: <latex>2^1 = 2 \not> 3</latex>, <latex>2^2 = 4 \not> 5</latex>, <latex>2^3 = 8 > 7</latex> ✓. By mathematical induction, <latex>2^n > 2n+1</latex> for all <latex>n \geq 3</latex>.</p>
Correct Answer: a