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. `G_(1)gtG_(2)gtG_(3)gt"......."`
B. `G_(1)ltG_(2)ltG_(3)lt"......."`
C. `G_(1)=G_(2)=G_(3)="......."`
D. `G_(1)ltG_(3)ltG_(5)lt"......." and "G_(2)gtG_(4)gtG_(6)gt"......."`

Answer 1

Correct Answer - C
`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))" and " H_(n)=(2A_(n-1)H_(n-1))/(A_(n-1)+H_(n-1))`
Clearly, `G_(1)=G_(2)=G_(3)="......"=sqrt(ab)`.