IIT-JEE 2007 · Paper 2 · Mathematics Q58

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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?

G_1>G_2>G_3>\cdots

G_1<G_2<G_3<\cdots

G_1=G_2=G_3=\cdots

G_1<G_3<G_5<\cdots

and

G_2>G_4>G_6>\cdots

Answer and worked solution

Official answer: C

Key identity. For two positive numbers, $AH=G^2$ . So $A_1H_1=G_1^2$ .

Next step. $A_2$ , $H_2$ are the A.M. and H.M. of $A_1$ and $H_1$ , so $G_2^2=A_2H_2=A_1H_1=G_1^2$ . Repeating, all the $G_n$ equal $G_1$ .

(C)

Topics:AM-GM-HM inequalities

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IIT-JEE 2007 · Paper 2, Mathematics question 58