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

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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
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

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
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.

已知1/a,1/b,1/c成等差数列
∴1/a+1/c=2/b,
∴bc+ab=2ac
（a-b/2）（c-b/2）=ac-ab/2-bc/2+bb/4=bb/4=(-b/2)(-b/2)
∴a-b/2,-b/2,c-b/2成等比数列

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