JEE Challenger
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MathMedAdv 2007 • Paper 2

Strict Monotonic Increase of Iterated Harmonic Means

Comprehension Passage

Let A1,G1,H1A_1, G_1, H_1 denote the arithmetic, geometric, and harmonic means, respectively, of two distinct positive numbers. For n≥2n \ge 2, let An−1A_{n-1} and Hn−1H_{n-1} have arithmetic, geometric, and harmonic means as An,Gn,HnA_n, G_n, H_n, respectively.

Which one of the following statements is correct?

Options

A

H1>H2>H3>…H_1 > H_2 > H_3 > \dots

B

H1<H2<H3<…H_1 < H_2 < H_3 < \dots

Correct
C

H1>H3>H5>…H_1 > H_3 > H_5 > \dots and H2<H4<H6<…H_2 < H_4 < H_6 < \dots

D

H1<H3<H5<…H_1 < H_3 < H_5 < \dots and H2>H4>H6>…H_2 > H_4 > H_6 > \dots

Topics & Concepts

Sequences and SeriesHarmonic MeanMonotonic SequencesAM-GM-HM Inequality

Step-by-Step Solution

Since An−1>Hn−1A_{n-1} > H_{n-1}, the harmonic mean HnH_n satisfies:
Hn=2An−1Hn−1An−1+Hn−1>2Hn−1Hn−1Hn−1+Hn−1=Hn−1H_n = \frac{2 A_{n-1} H_{n-1}}{A_{n-1} + H_{n-1}} > \frac{2 H_{n-1} H_{n-1}}{H_{n-1} + H_{n-1}} = H_{n-1}

Alternatively, since AnHn=G12A_n H_n = G_1^2 (a constant) and AnA_n is strictly decreasing (A1>A2>A3>…A_1 > A_2 > A_3 > \dots), Hn=G12AnH_n = \frac{G_1^2}{A_n} must be strictly increasing:
H1<H2<H3<…H_1 < H_2 < H_3 < \dots

Correct Answer: B

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