Let A_{1},G_{1} and H_{1} denote the arithmetic, geometric and harmonic means . respectively of two distinct positive numbers. For n ? 2 let A_{n - 1} and H_{n - 1} have . arithmetic, geometric and harmonic means as A_{n},G_{n},H_{n} respectively.: Sequence & Series (Mathematics)

Let

A_{1},G_{1}

and

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}

have . arithmetic, geometric and harmonic means as

A_{n},G_{n},H_{n}

respectively.

G_{1} > G_{2} > G_{3} > \ldots

G_{1} < G_{2} < G_{3} < \ldots

G_{1} = G_{2} = G_{3}\ldots

G_{1} < G_{3} < G_{5}\ldots

and

G_{2} > G_{4} > G_{6}\ldots

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Correct Answer:

Explanation

(c) : $A_1=\frac{a+b}{2}, G_1=\sqrt{a b}$ and $H_1 \\=\frac{2 a b}{a+b}$ Also, $A_n=\frac{\stackrel{2}{A}_{n-1}+H_{n-1}}{2}, \\G_n=\sqrt{A_{n-1} \cdot H_{n-1}}$ and $H_n \\=\frac{2 A_{n-1} H_{n-1}}{A_{n-1}+H_{n-1}}$ . $\\$ Now, $G_n{ }^2=A_n H_n=A_{n-1} H_{n-1}$ $\begin{aligned}& \therefore A_n H_n=A_{n-1} H_{n-1}=A_{n-2} H_{n-2}=\cdots=A_1 H_1 \\& \therefore G_1{ }^2=G_2{ }^2 \cdots =G_n{ }^2=a b \\& \text { or } G_1=G_2=G_3=\cdots=G_n=\sqrt{a b} .\end{aligned}$

Explanation

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