Question

If `A_(1)`, `A_(2)`, `A_(3)` , `G_(1)`, `G_(2)`, `G_(3)` , and `H_(1)`, `H_(2)`, `H_(3)` are the three arithmetic, geometric and harmonic means between two positive numbers `a` and `b(a gt b)`, then which of the following is/are true ?
A. `2G_(1)G_(3)=H_(2)(A_(1)+A_(3))`
B. `A_(2)H_(2)=G_(2)^(2)`
C. `A_(2)G_(2)=H_(2)^(2)`
D. `2G_(1)A_(1)=H_(1)(A_(1)+A_(3))`

Answer 1

Correct Answer - A::B
`(a,b)` We have `A_(1)=(3a+b)/(4)`, `A_(2)=(a+b)/(2)`, `A_(3)=(a+3b)/(4)`
`G_(1)=(a^(3)b)^(1//4)`, `G_(2)=(ab)^(1//2)`, `G_(3)=(ab^(3))^(1//4)`
`H_(1)=(4ab)/((a+3b))`, `H_(2)=(2ab)/((a+b))`, `H_(3)=(4ab)/((3a+b))`
`impliesA_(2)H_(2)=ab=G_(2)^(2)`
`G_(2)^(2)=A_(1)H_(3)=A_(2)H_(2)=A_(3)H_(1)=ab`