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>H</span><sub>1</sub> > <span>H</span><sub>2</sub> > <span>H</span><sub>3</sub> > ...</p>
<p>(B) <span>H</span><sub>1</sub> < <span>H</span><sub>2</sub> < <span>H</span><sub>3</sub> < ...</p>
<p>(C) <span>H</span><sub>1</sub> > <span>H</span><sub>3</sub> > <span>H</span><sub>5</sub> > ... and <span>H</span><sub>2</sub> < <span>H</span><sub>4</sub> < <span>H</span><sub>6</sub> < ...</p>
<p>(D) <span>H</span><sub>1</sub> < <span>H</span><sub>3</sub> < <span>H</span><sub>5</sub> < ... and <span>H</span><sub>2</sub> > <span>H</span><sub>4</sub> > <span>H</span><sub>6</sub> > ...</p>
Step-by-Step Solution
Key Concept: Since the harmonic mean of two distinct numbers is always less than their arithmetic mean, iterating the process forces the harmonic mean sequence to increase monotonically while remaining bounded by the constant geometric mean.
<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 harmonic mean is: \[H_2 = \frac{2A_1 H_1}{A_1 + H_1} > \frac{2H_1 H_1}{H_1 + H_1} = H_1\]</p><p>By the same reasoning, <span>A</span><sub>2</sub> > <span>H</span><sub>2</sub>, so <span>H</span><sub>3</sub> > <span>H</span><sub>2</sub>. The sequence <span>H</span><sub>n</sub> is strictly increasing and bounded above by the geometric mean <span>G</span>, so <span>H</span><sub>1</sub> < <span>H</span><sub>2</sub> < <span>H</span><sub>3</sub> < ...</p><p>∴ Answer is (B).</p>
Correct Answer: B