Q39 / 66 · JEE Advanced 2007 (Paper 2) · Mathematics
Paragraph:
Let $A_1, G_1, H_1$ denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For $n \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n, G_n, H_n$, respectively.Question:
Which one of the following statements is correct?
- A$G_1>G_2>G_3>\ldots$
- B$G_1 < G_2 < G_3 < \ldots$
- Correct answerC$G_1=G_2=G_3=\ldots$
- D$G_1 < G_3 < G_5 < \ldots$ and $G_2>G_4>G_6>\ldots$
Solution
$A_1=\frac{a+b}{2}, G_1=\sqrt{a b}$ and $H_1=\frac{2 a b}{a+b}$ $\begin{aligned} A_n=\frac{A_{n-1}+H_{n-1}}{2}, G_n & =\sqrt{A_{n-1} H_{n-1}}, \\ H_n & =\frac{2 A_{n-1} H_{n-1}}{A_{n-1}+H_{n-1}}\end{aligned}$
Clearly, $\quad G_1=G_2=G_3=\ldots=\sqrt{a b}$.
Hence, the correct answer is option: 'C'.