    Question

# Let A1,G1 and H1 denote the arithmetic, geometric and harmonic means respectively of two distinct positive numbers. For n≥2, let An−1 and Hn−1 have arithmetic, geometric and harmonic means as An,Gn,Hn respectively.Which of the following statements is correct?

A
G1>G2>G3>.....
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B
G1<G2<G3<.........
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C
G1=G2=G3........
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D
G1<G3<G5.... and G2<G4<G6.....
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Solution

## The correct option is C G1=G2=G3........Let 'a' and 'b' be two distinct positive numbers. Then,A1=a+b2, G1=√ab, H1=2aba+bIt is given that for n≥2An=An−1+Hn−12, Gn=√An+Hn−1, Hn=2An−1Hn−1An−1+Hn+1∴AnHn=An−1Hn−1Gn+1=√AnHn=√An−1Hn−1=Gn for n≥2∴Gn=Gn−1=Gn−2=…..=G3=G2=G1=√ab.   Suggest Corrections  0      Similar questions  Related Videos   Relation between AM, GM and HM
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