Sequences & Series
Arithmetic, Geometric and Harmonic Means
Grade 11
Question:
<p>Let <span>A</span><sub>1</sub>, <span>G</span><sub>1</sub>, <span>H</span><sub>1</sub> denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For <span>n</span> ≥ 2, let <span>A</span><sub>n–1</sub> and <span>H</span><sub>n–1</sub> have arithmetic, geometric and harmonic means as <span>A</span><sub>n</sub>, <span>G</span><sub>n</sub>, <span>H</span><sub>n</sub> respectively. Which one of the following statements is correct?</p>
<p>(A) <span>A</span><sub>1</sub> > <span>A</span><sub>2</sub> > <span>A</span><sub>3</sub> > ...</p>
<p>(B) <span>A</span><sub>1</sub> < <span>A</span><sub>2</sub> < <span>A</span><sub>3</sub> < ...</p>
<p>(C) <span>A</span><sub>1</sub> > <span>A</span><sub>3</sub> > <span>A</span><sub>5</sub> > ... and <span>A</span><sub>2</sub> < <span>A</span><sub>4</sub> < <span>A</span><sub>6</sub> < ...</p>
<p>(D) <span>A</span><sub>1</sub> < <span>A</span><sub>3</sub> < <span>A</span><sub>5</sub> < ... and <span>A</span><sub>2</sub> > <span>A</span><sub>4</sub> > <span>A</span><sub>6</sub> > ...</p>
Step-by-Step Solution
Key Concept: The AM-HM inequality ensures that the arithmetic mean at each step is strictly greater than the harmonic mean, causing the sequence of arithmetic means to decrease monotonically.
<p><strong>Analysis:</strong> For two numbers with arithmetic mean <span>A</span> and harmonic mean <span>H</span>, by the AM-HM inequality: <span>A</span> ≥ <span>H</span> with equality only when the numbers are equal. Since we have distinct positive numbers, <span>A</span><sub>1</sub> > <span>H</span><sub>1</sub>.</p><p>The new arithmetic mean is: \[A_2 = \frac{A_1 + H_1}{2} < \frac{A_1 + A_1}{2} = A_1\]</p><p>By the same reasoning, <span>A</span><sub>2</sub> > <span>H</span><sub>2</sub>, so <span>A</span><sub>3</sub> < <span>A</span><sub>2</sub>. The sequence <span>A</span><sub>n</sub> is strictly decreasing and bounded below by the geometric mean <span>G</span>, so <span>A</span><sub>1</sub> > <span>A</span><sub>2</sub> > <span>A</span><sub>3</sub> > ...</p><p>∴ Answer is (A).</p>
Correct Answer: A