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Let A1 , G1, H1denote the arithmetic, g...

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

`G_(1) gt G_(2) gt G_(3) gt ...`

B

`G_(1) lt G_(2) lt G_(3) lt...`

C

`G_(1) = G_(2) = G_(3) = ...`

D

`G_(1) lt G_(3) lt G_(5) lt .... And G_(2) gt G_(4) gt G_(6) gt ...`

Text Solution

Verified by Experts

The correct Answer is:
C
Let a and b are two numbers. Then,
`A_(1) = (a +b)/(2) , G_(1) = sqrt(ab), H_(1) = (2ab)/(a +b)`
`A_(n) = (A_(n-1) + H_(n-1))/(2)`
`G_(n) = sqrt(A_(n-1) H_(n-1))`,
`H_(n) = (2A_(n-1) H_(n-1))/(A_(n-1) + H_(n-1))`
Clearly, `G_(1) = G_(2) = G_(3) = ...= sqrt(ab)`
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