若b是a與c的等差中項,y是x與z的等比中項,且x,y,z都是正數,求證(b-c)logmX+(c-a)logmY+(a-b)logmZ=0,其中m大于0,且m不等于1

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∵b是a與c的等差中項,∴可設b=a+d,c=a+2d(d是a,b,c的公差).又∵Y是X與Z的等比中項,∴Y^2=XZ.而x,y,z都是正數,∴Y^(2d)=(X^d)(Z^d).∴[Y^(2d)][X^(-d)][Z^(-d)]=1,即{X^[(a+d)-(a+2d)]}{Y^[(a+2d)-a]}{Z^[a-(a+d)]}=1,∴[X^(b-c)][Y^(c-a)][Z^(a-b)]=1.又∵m大于0,且m不等于1,∴logm{[X^(b-c)][Y^(c-a)][Z^(a-b)]}=logm1=0,即logm[X^(b-c)]+logm[Y^(c-a)]+logm[Z^(a-b)]=0,∴(b-c)logmX+(c-a)logmY+(a-b)logmZ=0.