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If x,y,z are in AP . Prove that (x+2y-z)(2y+z-x)(z+x-y)=4xyz

If x,y,z are in AP . Prove that (x+2y-z)(2y+z-x)(z+x-y)=4xyz

Grade:11

2 Answers

Harshit Panwar
108 Points
6 years ago
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
Hence proved.
ARKADEEP DAS
44 Points
6 years ago
hi suraj
this is typical application of ideas
take x as starter of the series and common difference to be d
then y=x+d
z=x+2d
therefore (x+2y-z)=(x+2x+2d-x-2d)=2x
(2y+z-x)=(2x+2d+x+2d-x)=2(x+2d)=2z
(z+x-y)=(x+2d+x-x-d)=y
so product gives u 4xyz
 

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