JEE Challenger
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MathMedAdv 2007 • Paper 2

Strict Monotonic Decrease of Iterated Arithmetic 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

A1>A2>A3>…A_1 > A_2 > A_3 > \dots

Correct
B

A1<A2<A3<…A_1 < A_2 < A_3 < \dots

C

A1>A3>A5>…A_1 > A_3 > A_5 > \dots and A2<A4<A6<…A_2 < A_4 < A_6 < \dots

D

A1<A3<A5<…A_1 < A_3 < A_5 < \dots and A2>A4>A6>…A_2 > A_4 > A_6 > \dots

Topics & Concepts

Sequences and SeriesAM-GM-HM InequalityMonotonic SequencesSequences and Series

Step-by-Step Solution

By the classical AM-HM inequality for two distinct positive numbers An−1A_{n-1} and Hn−1H_{n-1}:
An−1>Hn−1A_{n-1} > H_{n-1}

By definition, AnA_n is the arithmetic mean of An−1A_{n-1} and Hn−1H_{n-1}:
An=An−1+Hn−12<An−1+An−12=An−1A_n = \frac{A_{n-1} + H_{n-1}}{2} < \frac{A_{n-1} + A_{n-1}}{2} = A_{n-1}

Therefore, An<An−1A_n < A_{n-1} for all n≥2n \ge 2, meaning the sequence of arithmetic means is strictly monotonically decreasing:
A1>A2>A3>…A_1 > A_2 > A_3 > \dots

Correct Answer: A

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