Let A_1,G_1 and H_1 denote the arithmetic, geometric and harmonic means . respectively of two distinct positive numbers. Forn >= 2 let A_n - 1 and H_n - 1 have . arithmetic, geometric and harmonic means as A_n,G_n,H_n respectively.

(c) : A_1=(a+b / 2), G_1=sqrt(a b) and H_1 =(2 a b / a+b) Also, A_n=fracstackrel2A_n-1+H_n-12, G_n=sqrtA_n-1 * H_n-1 and H_n =frac2 A_n-1 H_n-1A_n-1+H_n-1. Now, G_n ^2=A_n H_n=A_n-1 H_n-1 \ Therefore A_n H_n=A_n-1 H_n-1=A_n-2 H_n-2= * s=A_1 H_1 Therefore G_1 ^2=G_2 ^2 * s =G_n ^2=a b or G_1=G_2=G_3= * s=G_n=sqrt(a b) .\

Let A_{1},G_{1} and H_{1} denote the arithmetic, geomet... | 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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