Question

if x,y,z are in AP then find the value of (x+y-z) (-x+y+z)

Answer 1

X,Y,Z are in AP
then y-x=z-y
2y=x+z
y=x+z/2
According to question
(x+y-z)(-x+y+z)
x-z+x+z  (-x+x+z +z)
       2            2


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

1/4 (3x-z)(3z-x)

Answer 2

x,y,z...

d=common difference
x+d=y
x+2d=z

x+y=x+x+d
(x+y-z)=x+x+x+d-x+2d=3x+d-x+2d=2x-d
(-x+y+z)=-x+x+d+x+2d=x+3d
so
(x+y-z) (-x+y+z)=(2x-d) (x+3d)