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Relation between AM,GM,HM for 2 Numbers

Let A1,G1, ...

Question

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 $H_{n - 1}$ hav arithmetic, gemetric and hrmonic means as $A_{n}, G_{n}, H_{n}$ respectively.
Which one of the following statements is correct?

A

G_{1} > G_{2} > G_{3} > . . .

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B

G_{1} < G_{2} < G_{3} < . . .

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C

G_{1} = G_{2} = G_{3} = . . .

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D

G_{1} < G_{3} < G_{5} < . . .

and

G_{2} > G_{4} > G_{6} > . . .

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Solution

The correct option is D $G_{1} = G_{2} = G_{3} = . . .$
$A_{1} = \frac{a + b}{2}; G_{1} = \sqrt{a b}; H_{1} = \frac{2 a b}{a + b} A_{n} = \frac{A_{n - 1} + H_{n - 1}}{2}; G_{n} = \sqrt{A_{n - 1} H_{n - 1}}; H_{n} = 2 \frac{A_{n - 1} H_{n - 1}}{A_{n - 1} + H_{n - 1}}$
$\Rightarrow G_{1} = G_{2} = G_{3} = \dots = \sqrt{a b}$

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