IIT-JEE 2007 · Paper 2 · Mathematics Q59

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MathematicsSingle correct+4 / -1Question 59 of 66

Passage

Paragraph for Question Nos. 58 to 60

Let $A_1$ , $G_1$ , $H_1$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n\ge2$ , let $A_{n-1}$ and $H_{n-1}$ have arithmetic, geometric and harmonic means as $A_n$ , $G_n$ , $H_n$ , respectively.

(The paragraph is not printed in the official archive copy (which reproduces only the questions with their answers); its text is restored from the JEE 2007 question paper.)

Which one of the following statements is correct?

A_1>A_2>A_3>\cdots

A_1<A_2<A_3<\cdots

A_1>A_3>A_5>\cdots

and

A_2<A_4<A_6<\cdots

A_1<A_3<A_5<\cdots

and

A_2>A_4>A_6>\cdots

Answer and worked solution

Official answer: A

Decreasing. Since the numbers are distinct, $H_{n-1}<A_{n-1}$ , so
$A_n=\frac{A_{n-1}+H_{n-1}}2<A_{n-1}.$
Thus $A_1>A_2>A_3>\cdots$ .

(A)

Topics:AM-GM-HM inequalities

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IIT-JEE 2007 · Paper 2, Mathematics question 59