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)= . . .andG_(2)gtG_(4)gtG_(6)gt . . .`

Answer 1

Correct Answer - C
Let a and b be two distinct positive numbers.
Then,
`A_(1)=(a+b)/(2),G_(1)sqrt(ab)andH_(1)=(2ab)/(a+b)`
It is given that for `nge2`
`A_(n)=(A_(n-1)+H_(n-1))/(2),G_(n)=sqrt(A_(n-1)H_(n-1))`
`andH_(n)=(2A_(n-1)H_(n-1))/(A_(n-1)+H_(n-1))`
`rArr" "A_(n)H_(n)=A_(n-1)H_(n-1)`
`:." "G_(n+1)=sqrt(A_(n)H_(n))=sqrt(A_(n-1)H_(n-1))=G_(n)"for "nge2`
`rArr" "G_(n)=G_(n-1)=G_(n-2)= . . .=G_(3)=G_(2)=G_(1)=sqrt(ab)`.