Question

已知(b-c)log_mx+(c-a)log_my+(a-b)log_mz=0,

(1)若a、b、c為公差d≠0的等差數(shù)列.求證:x、y、z成等比數(shù)列;

(2)若x、y、z為公比q≠1的等比數(shù)列.求證:a、b、c成等差數(shù)列.

Answer 1

證明:由條件式知x、y、z都為正數(shù),

且x^b-c·y^c-a·z^a-b=1,                                                           (*)

(1)若a、b、c成等差數(shù)列,公差d≠0,

則(*)變?yōu)閤^-d·y^2d·z^-d=1

(x^-1y²z^-1)^d=1(xz)^-1·y²=1

y²=xz(x、y、z都為正數(shù)),

∴x、y、z成等比數(shù)列.

(2)若x、y、z成等比數(shù)列,公比q≠1,

則y=xq,z=xq²,代入(*)得x^b-c·(xq)^c-a·(xq²)^a-b=1,

化簡為q^c+a-2b=12b=a+c,

∴a、b、c成等差數(shù)列.