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>G</span><sub>1</sub> > <span>G</span><sub>2</sub> > <span>G</span><sub>3</sub> > ...</p>
<p>(B) <span>G</span><sub>1</sub> < <span>G</span><sub>2</sub> < <span>G</span><sub>3</sub> < ...</p>
<p>(C) <span>G</span><sub>1</sub> = <span>G</span><sub>2</sub> = <span>G</span><sub>3</sub> = ...</p>
<p>(D) <span>G</span><sub>1</sub> < <span>G</span><sub>3</sub> < <span>G</span><sub>5</sub> < ... and <span>G</span><sub>2</sub> > <span>G</span><sub>4</sub> > <span>G</span><sub>6</sub> > ...</p>
Step-by-Step Solution
Key Concept: The geometric mean of the arithmetic and harmonic means of two numbers equals the geometric mean of the original numbers. This invariant property holds at each iteration.
<p><strong>Analysis:</strong> For two numbers <span>a</span> and <span>b</span>, the geometric mean <span>G</span> = \(\sqrt{ab}\). When we compute the arithmetic mean <span>A</span> = \(\frac{a+b}{2}\) and harmonic mean <span>H</span> = \(\frac{2ab}{a+b}\), and then find the geometric mean of <span>A</span> and <span>H</span>, we get:</p><p>\[G' = \sqrt{A \cdot H} = \sqrt{\frac{a+b}{2} \cdot \frac{2ab}{a+b}} = \sqrt{ab} = G\]</p><p>Therefore, <span>G</span><sub>n</sub> = <span>G</span><sub>n–1</sub> for all <span>n</span> ≥ 2, which means <span>G</span><sub>1</sub> = <span>G</span><sub>2</sub> = <span>G</span><sub>3</sub> = ...</p><p>∴ Answer is (C).</p>
Correct Answer: C