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IIT JEE Advanced 2007 Paper 2

Paragraph Type

This section contains 2 paragraphs. Based on the paragraph, there are 3 questions. Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct.

Paragraph for Questions 58–60: Let

A_1, G_1, 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}

has arithmetic, geometric and harmonic means as

A_n, G_n, H_n

, respectively.

Section: Mathematics

Question : 58 of 66

Marks: +1, -0

Which one of the following statements is correct?

[JEE Adv 2007 P2]

$G_1$ > $G_2$ > $G_3$ > …
$G_1$ < $G_2$ < $G_3$ < …
$G_1$ = $G_2$ = $G_3$ = …
$G_1$ < $G_3$ < $G_5$ < … and $G_2$ > $G_1$ > $G_6$ > …

Validate

Solution:

Let us consider

A_1

\frac{a+b}{2}

G_1

\sqrt{ab}

H_1

\frac{2ab}{a+b}

A_n

\frac{A_{n-1}+H_{n-1}}{2}

G_n

\sqrt{A_{n-1}.H_{n-1}}

H_n

\frac{2A_{n-1}.H_{n-1}}{A_{n-1}+H_{n-1}}

Therefore,

G_2

\sqrt{A_1 . H_1}

\sqrt{ab}

Thus, it is clear that

G_1

G_2

G_3

= ... =

\sqrt{ab}

and hence we conclude that option (C) is the correct one.

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