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Let`A_(1),G_(1),H_(1)` denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For `nge2`, let `A_(n-1),G_(n-1)" and "H_(n-1)` has arithmetic,geometric and harmonic means as `A_(n),G_(n),H_(n)` respectively.
Which of the following statement is correct?
a. `H_(1)gtH_(2)gtH_(3)gt"......."`
b. `H_(1)ltH_(2)ltH_(3)lt"......."`
c. `H_(1)gtH_(3)gtH_(5)gt"......." " and " H_(2)ltH_(4)ltH_(6)lt"......."`
d. `H_(1)ltH_(3)ltH_(5)lt"......."" and " H_(2)gtH_(4)gtH_(6)gt"......."`

A

`H_(1)gtH_(2)gtH_(3)gt"......."`

B

`H_(1)ltH_(2)ltH_(3)lt"......."`

C

`H_(1)gtH_(3)gtH_(5)gt"......." " and " H_(2)ltH_(4)ltH_(6)lt"......."`

D

`H_(1)ltH_(3)ltH_(5)lt"......."" and " H_(2)gtH_(4)gtH_(6)gt"......."`

Text Solution

Verified by Experts

The correct Answer is:
B
As above`A_(1)gtH_(2)gtH_(1),A_(1)gtH_(3)gtH_(2)`
`:.H_(1)ltH_(2)ltH_(3)lt"......"`
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Let A_1 , G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n >2, let A_(n-1),G_(n-1) and H_(n-1) has arithmetic, geometric and harmonic means as A_n, G_N, H_N, respectively a. A_(1)gtA_(2)gtA_(3)gt"......." b. A_(1)ltA_(2)ltA_(3)lt"......." c. A_(1)gtA_(3)gtA_(5)gt"......."" and " A_(2)ltA_(4)ltA_(6)lt"......." d. A_(1)ltA_(3)ltA_(5)lt"......."" and " A_(2)gtA_(4)gtA_(6)gt"......."

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