已知等差数列的相邻三项为1/a,1/b,1/c,求证(b+c)/a,(c+a)/b,(a+b)/c也成等差数列
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已知等差数列的相邻三项为1/a,1/b,1/c,求证(b+c)/a,(c+a)/b,(a+b)/c也成等差数列
已知等差数列的相邻三项为1/a,1/b,1/c,求证(b+c)/a,(c+a)/b,(a+b)/c也成等差数列
已知等差数列的相邻三项为1/a,1/b,1/c,求证(b+c)/a,(c+a)/b,(a+b)/c也成等差数列
由1/a,1/b,1/c
得1/a+1/c=2/b
简化后 2ac=ab+bc
2abc=ab^2+b^2c
2abc+a^2b+bc^2=ab^2+b^2c+a^2b+bc^2
(ab+bc)(c+a)=b(bc+c^2+a^2+ab)
bc+c^2+a^2+ab c+a
-------------=---
ab+bc b
bc+c^2+a^2+ab c+a
-------------=---
2ac b
b+c a+b c+a
--- + --- = 2---
a c b
(b+c)/a,(c+a)/b,(a+b)/c也成等差数列
等差数列的相邻三项为1/a,1/b,1/c
2/b=(1/a)+(1/b)
所以:
(b+c)/a +(a+b)/c
=(b/a)+(c/a)+(a/c)+(b/c)
=b((1/a)+(1/c))+(c/a)+(a/c)
=b*(2/b)+(c/a)+(a/c)
=2+(c/a)+(a/c)
=(1+(c/a))+(1+(a/c))...
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等差数列的相邻三项为1/a,1/b,1/c
2/b=(1/a)+(1/b)
所以:
(b+c)/a +(a+b)/c
=(b/a)+(c/a)+(a/c)+(b/c)
=b((1/a)+(1/c))+(c/a)+(a/c)
=b*(2/b)+(c/a)+(a/c)
=2+(c/a)+(a/c)
=(1+(c/a))+(1+(a/c))
=(a+c)/a + (a+c)/c
=(a+c)((1/a)+(1/c))
=(a+c)(2/b)
=2*((a+c)/b)
所以:(b+c)/a,(c+a)/b,(a+b)/c也成等差数列
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