Question

Show that `x^2+x y+y^2,z^2+xz+x^2, y^2+y z+z^2,` are consecutive terms of an A.P., if `x ,y and z` are in A.P.

Answer 1

`y-x=z-y-(1)`
`(x^2+xy+y^2,z^2+zx+x^2,y^2+yz+z^2)`
`t_2-t_1=z^2+zx+x^2-x^2-xy-y^2`
`=(z^2-y^2)+x(z-y)`
`=(z-y)[z+y+x]`
`=(z-y)(x+y+z)-(2)`
`t_3-t_2=y^2+yz+z^2-z^2-zx-x^2`
`=y^2-x^2+z(y-x)`
`=(y-x)[y+x+z]`
`=(z-y)(x+y+z)-(3)`
Using equation 2 and 3
`t_2-t_1=t_3-t_2`
`t_1.t_2.t_3` are in AP.