IIT-JEE 2007 · Paper 2 · Mathematics Q60

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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?

H_1>H_2>H_3>\cdots

H_1<H_2<H_3<\cdots

H_1>H_3>H_5>\cdots

and

H_2<H_4<H_6<\cdots

H_1<H_3<H_5<\cdots

and

H_2>H_4>H_6>\cdots

Answer and worked solution

Official answer: B

Increasing. $H_n$ is the harmonic mean of $A_{n-1}$ and $H_{n-1}$ , which lies strictly between them, so $H_n>H_{n-1}$ . (Equivalently $H_n=\dfrac{G_1^2}{A_n}$ with $A_n$ decreasing.)

So $H_1<H_2<H_3<\cdots$ .

(B)

Topics:AM-GM-HM inequalities

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IIT-JEE 2007 · Paper 2, Mathematics question 60