(b-c)lnx+(c-a)lny+(a-b)lnz=0,且x,y,z构成等比数列,求证a,b,c为等差数列

题目

(b-c)lnx+(c-a)lny+(a-b)lnz=0,且x,y,z构成等比数列,求证a,b,c为等差数列

答案

x,y,z构成等比数列,则y²=xz,
2lny=lnx+lnz,
即lnx-2lny+lnz=0,
对比(b-c)lnx+(c-a)lny+(a-b)lnz=0,
设b-c=t,
则a-b=t,
所以a,b,c为等差为-t的等差数列.t∈R.