Question

Let $A_1,G_1$ and $H_1$ denote the arithmetic, geometric and harmonic means respectively of two distinct positive numbers. For $n\ge 2$, let $A_{n-1}$ and $H_{n-1}$ have arithmetic, geometric and harmonic means as $A_n,G_n,H_n$ respectively.
Which of the following statements is correct?

• A. $G_1>G_2>G_3>.....$
• B. $G_1<G_2<G_3<.........$
• C. $G_1=G_2=G_3........$
• D. $G_1<G_3<G_5....$ and $G_2<G_4<G_6.....$

Answer 1

The correct option is C $G_1=G_2=G_3........$

Let 'a' and 'b' be two distinct positive numbers. Then,

$A_1=\frac{a+b}{2}$, $G_1=\sqrt{ab}$, $H_1=\frac{2ab}{a+b}$

It is given that for $n\ge 2$

$A_n=\frac{A_{n-1}+H_{n-1}}{2}$, $G_n=\sqrt{A_n+H_{n-1}}$, $H_n=\frac{2A_{n-1}{H}_{n-1}}{A_{n-1}+H_{n+1}}$

$\therefore A_n{H}_n=A_{n-1}{H}_{n-1}$

$G_{n+1}=\sqrt{A_n{H}_n}=\sqrt{A_{n-1}{H}_{n-1}}=G_n$ for $n\ge 2$

$\therefore G_n=G_{n-1}=G_{n-2}=\dots ..=G_3=G_2=G_1=\sqrt{ab}$.