Question

若三个正数a,b,c成等比数列,三数x,y ,.z 成等差数列,则（y-z)lga+(z-x)lgb+(x-y)lgc=

Answer 1

由题意： b^2 =ac , 2y = x + z 即：y-z = x-y

原式=lga^(y-z) + lgb^(z-x) + lgc^(x-y)

=lga^(x-y) + lgb^(z-x) + lgc^(x-y)

=lg[a^(x-y) × c^(x-y)] + lgb^(z-x)

=lg[(ac)^(x-y)] + lgb^(z-x)

=lg[(b^2)^(x-y)] + lgb^(z-x)

=lgb^(2x-2y) + lgb^(z-x)

=lgb^(2x-2y+z-x)

=lgb^(x-2y+z)

=lgb^[x-(x+z)+z]

=lgb^0

=lg1

=0