If `a,b,c,d` be four distinct positive quantities in HP, then
(a) `a+dgtb+c`
(b)`adgt bc`

Explanation

`therefore a,b,c,d` are in HP.
(a) Applying AMgtHM
For first three members,
`(a+c)/(2)gtb`
`implies a+cgt2b" " "….(ix)"`
For last three members, `(b+d)/(2)gtc`
`implies b+dgt2c " " "….(x)"`
From Eqs. (ix) and (x), we get
`a+c+b+dgt2b+2c`
or `a+dgtb+c`
(b) Applying GMgt HM
For first three members, `sqrt(ac)gtb`
`implies acgtb^(2) " " "....(xi)"`
For last three members, `sqrt(bd)gtc`
`implies bdgtc^(2) " " "....(xii)"`
From Eqs. (xi) and (xii), we get
`(ac)(bd)gtb^(2)c^(2)`
or `adgtbc`