For n ∈ N , let A n , G n and H n be the arithmetic, geometric and harmonic means of n + 1 , n + 2 , … , 2 n , respectively. Find lim n → ∞ A n / n .

A n = [( n + 1 ) + 2 n ] /2 = ( 3 n + 1 ) /2 . Therefore A n / n → 3/2 .

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MathematicsSequence & SeriesMeans / inequalityMedium2 minQB

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For $n \in N$ , let $A_{n}, G_{n}$ and $H_{n}$ be the arithmetic, geometric and harmonic means of $n + 1, n + 2, \dots, 2 n$ , respectively. Find $lim_{n \to \infty} A_{n} / n$ .

Options:

Answer:

Solution:

$A_{n} = [(n + 1) + 2 n] /2 = (3 n + 1) /2$ . Therefore $A_{n} / n \to 3/2$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Means / inequality

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