已知1/a,1/b,1/c成等差数列,求证a－b/2,b/2,c－b/2成等比数列

Question

已知1/a,1/b,1/c成等差数列,求证a－b/2,b/2,c－b/2成等比数列

Answer 1

证：
1/a,1/b,1/c成等差数列,则2/b=1/a +1/c
2/b=(a+c)/(ac)
b(a+c)=2ac
a+c=2ac/b
(a- b/2)(c- b/2)
=ac- ab/2 -bc/2 +b^2/4
=ac-(b/2)(a+c)+b^2/4
=ac -(b/2)(2ac/b)+b^2/4
=b^2/4
(b/2)^2=b^2/4
(a- b/2)(c- b/2)=(b/2)^2
a- b/2,b/2,c- b/2成等比数列.