(A)a,b,c,d are in AP " " [a,b,c,d are positive real numbers ]
By `AMgtGM` for a,b,c
`bgtsqrt(ac)`
` implies b^(2)gtac " " "……(i)"`
Now, applying for b,c,d
`cgtsqrt(bd) impliesc^(2)gtbd" " "……(ii)"`
From Eq.(i) and (ii), we get
`b^(2)c^(2)gt(ac)(bd) implies bcgtad`
Again, applying `AMgtHM` for a,b,c
`bgt(2)/((1)/(a)+(1)/(c)) implies (1)/(a)+(1)/(c)gt(2)/(b) " " "......(iii)"`
For last 3 terms b,c,d
`cgt(2)/((1)/(b)+(1)/(d)) implies (1)/(b)+(1)/(d)gt(2)/(c) " " "......(iv)"`
From Eqs. (iii) and (iv), we get
`(1)/(a)+(1)/(c)+(1)/(b)+(1)/(d)gt(2)/(b)+(2)/(c)`
`implies (1)/(a)+(1)/(d)gt(1)/(b)+(1)/(c)`.
(B) a,b,c,d are in GP.
For a,b,c applying `AMgtGM`,
`implies (a+c)/(2)gtb implies a+cgt2b" " ".....(i)"`
Similarly, for b,c,d
`b+dgt2c" " ".....(ii)"`
From Eqs. (i) and (ii), we get
`a+b+c+dgt2b+2c implies a+dgtb+c`
Now, applying `GMgtHM` for a,b,c
`bgt(2ac)/(a+c)`
` implies(1)/(c)+(1)/(a)gt(2)/(c)" " ".......(iii)"`
Similarly, for b,c,d we get
`(1)/(d)+(1)/(b)gt(2)/(c)" " ".......(iv)"`
On adding Eqs. (iii) and (iv), we get
`(1)/(a)+(1)/(b)+(1)/(c)+(1)/(d)gt2((1)/(b)+(1)/(c))`
`implies (1)/(a)+(1)/(d)gt(1)/(b)+(1)/(c)`.
(C ) a,b,c,d are in HP.
Applying `AMgtHM` for a,b,c
`(a+c)/(2)gtb`
`implies a+cgt2b" " "......(i)"`
Similarly,for last 3 terms b,c,d
`b+dgt2c" " "......(ii)"`
On addding Eqs. (i) and (ii), we get
`a+b+c+dgt2b+2c`
`implies a+dgtb+c`
Again, applying `GMgtHM` for a,b,c
`sqrt(ac)gtb`
`impliesacgtb^(2)" " ".......(iii)"`
Similarly, for b,c,d
`implies bdgtc^(2) " " "......(iv)"`
On multiplying Eqs. (iii) and (iv), we get
`abcdgtb^(2)c^(2)`
`adgtbc`.
