Sequences & Series
AM-GM Inequality
Grade 11
Question:
<p>The product of n positive numbers is unity. Their sum is</p>
<p>(a) a positive integer</p>
<p>(b) equal to \(n + \frac{1}{n}\)</p>
<p>(c) divisible by n</p>
<p>(d) never less than n</p>
Step-by-Step Solution
Key Concept: Apply AM-GM inequality to show that the sum of positive numbers with product 1 is at least n.
<p><strong>Solution:</strong> By the AM-GM inequality, for n positive numbers $a_1, a_2, \ldots, a_n$ with product equal to 1:</p><p>$$\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n} = \sqrt[n]{1} = 1$$</p><p>Therefore, $a_1 + a_2 + \cdots + a_n \geq n$, so the sum is never less than n.</p><p>∴ Answer is (d).</p>
Correct Answer: D