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 . . .`

Let `A_1 , G_1, H_1`denote the arithmetic, geometric and harmonic mean

1.	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» 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)`.

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