If x,y,z are in A.P., show that (x+2y-z)(2y+z-x)(z+x-y)=4xyz

Since x,y,z are in AP.

Therefore,

d=y-x

=z-y

=(z-x)/2

(x+2y-z)

=x+y+y-z

=x+y-(z-y)

=x+y-(y-x)

=x+y-y+x

= 2x

(2y+z-x)=

2y+2z-2y

=2z.

{ (z-x)/2=z-y => z-x=2z-2y }

(z+x-y)

=z-(y-x)

=z-(z-y)

=z-z+y

=y

(x+2y-z)(2y+z-x)(z+x-y)

=2x ? 2z ? y

=4xyz