If `a,b,c,d` be four distinct positive quantities in AP, then
(a) `bcgtad`
(b) ` c^(-1)d^(-1)+a^(-1)b^(-1)gt2(b^(-1)d^(-1)+a^(-1)c^(-1)-a^(-1)d^(-1))`

Explanation

`therefore a,b,c,d` are in AP.
(a) Applying AMgtGM
For first three members, `bgtsqrt(ac)`
`implies b^(2)gt ac " " "….(i)"`
and for last three members, `cgtsqrt(bd)`
`implies c^(2)gt bd " " "……(ii)"`
From Eqs. (i) and (ii), we get
`b^(2)c^(2)gt(ac)(bd)`
Hence, `bcgtad`
(b) Applying AMgtHM
For first three members,
`bgt(2ac)/(a+c)`
`implies ab+bcgt2ac " " ".....(iii)"`
For last three members, `cgt(2bd)/(b+d)`
`bc+cdgt2bd " " "...(iv)"`
From Eqs. (iii) and (iv), we get
`ab+bc+bc+cdgt2ac+2bd`
or `ab+cdgt2(ac+bd-bc)`
Dividing in each term by `abcd`, we get
` c^(-1)d^(-1)+a^(-1)b^(-1)gt2(b^(-1)d^(-1)+a^(-1)c^(-1)-a^(-1)d^(-1))`