已知（b-c)LOGmX+(c-a)LOGmY+(a-b)LOGmZ=o,设x,y,z成等比数列,且公比不为1,求证x,z,y成等差数列.

Question

已知（b-c)LOGmX+(c-a)LOGmY+(a-b)LOGmZ=o,设x,y,z成等比数列,且公比不为1,求证x,z,y成等差数列.

Answer 1

（b-c)LOGmX+(c-a)LOGmY+(a-b)LOGmZ=0
LOGmX^(b-c)+LOGmY^(c-a)+LOGmZ^(a-b)=0
X^(b-c)*Y^(c-a)*Z^(a-b)=1
X^(b-c)*X^(c/2-a/2)*Z^(c/2-a/2)*Z^(a-b)=1
X^(b-c/2-a/2)*Z^(c/2+a/2-b)=1
(x/z)^(b-c/2-a/2)=1
若x=z,q=1,不可
故b-c/2-a/2=0
2b=a+c
a,b,c成等差数列