Question

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

Answer 1

Here is ur answer.....

Given, x,y and z are in AP

Therefore they have a common difference

so, y - x = z - y

==> y + y = z + x

==> 2y = z + x

To prove ( x + 2y - z ) ( 2y + z - x ) ( z + x - y ) = 4xyz

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

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

=( 2x ) ( 2z ) ( y )

= 4xyz = RHS

Hence proved...

Hope it helps!☺☺

Answer 2

Step-by-step explanation:

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.

hope this helps you ❤️