若x,y,z成等差数列,则(z-x)^2-4(x-y)(y-z)=_________

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若x,y,z成等差数列,则(z-x)^2-4(x-y)(y-z)=_________
若x,y,z成等差数列,则(z-x)^2-4(x-y)(y-z)=_________

若x,y,z成等差数列,则(z-x)^2-4(x-y)(y-z)=_________
设 x,y,z成等差数列,公差等于d
x+d=y y+d=z x+2d=z
(z-x)^2=(2d)^2=4d^2
4(x-y)(y-z)=4(-d)(-d)=4d^2
原式=4d^2-4d^2=0

(z-x)^2-4(x-y)(y-z)
=4d^2-4(-d)(-d)
=4d^2-4d^2
=0

设公差为d
原式=(2d)^2-4(-d)(-d)
=4d^2-4d^2
=0

y=X+n , z=x+2n 则原式=(2n)^2-4(-n)(-n)=4n-4n=0