Question

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) gt A_(2) gt A_(3) gt ...`
B. `A_(1) lt A_(2) lt A_(3) lt ...`
C. `A_(1) gt A_(3) gt A_(5) gt .... and A_(2) lt A_(4) lt A_(6) lt...`
D. `A_(1) lt A_(3) lt A_(5)lt ....and A_(2) gt A_(4) gt A_(6) gt ...`

Answer 1

Correct Answer - A
`A_(2)` is AM of `A_(1) and H_(1) and A_(1) gt H_(1)`
`rArr A_(1) gt A_(2) gt H_(1)`
`A_(3)` is AM of `A_(2) and H_(2) and A_(2) gt H_(2)`
`{:(rArr,A_(2),gt,A_(3),gt,H_(2),),(,vdots,,vdots,,vdots,),( :.,A_(1),gt,A_(2),gt,A_(3),gt...):}`