If a,b,c are in HP, then prove that `(a+b)/(2a-b)+(c+b)/(2c-b)gt4`.

Explanation

Since, a,b,c are in HP.
`:. (2)/(b)=(1)/(a)+(1)/(c )" " "…….(i)"`
and let `P=(a+b)/(2a-b)+(c+b)/(2c-b)`
`=(a+(2ac)/(a+c))/(2a-(2ac)/(a+c))+(c+(2ac)/(a+c))/(2a-(2ac)/(a+c))" "[" from Eqs.(i) "]` ltbr. `=(a+3c)/(2a)+(3a+c)/(2c)=1+(3)/(2)((c)/(a)+(a)/(c))" " ".....(ii)"`
`:.AMgtGM " " " " [:.ane c]`
`:.((c)/(a)+(a)/(c))gt2`
`implies (3)/(2) ((c)/(a)+(a)/(c))gt3`
or `1+(3)/(2)((c)/(a)+(a)/(c))gt1+3` or `Pgt4`
Hence, `(a+b)/(2a-b)+(c+b)/(2c-b)gt4`.