Question

cho \(\frac{1}{c}=\frac{1}{2}.\left(\frac{1}{a}+\frac{1}{b}\right)\)

cmr:\(\frac{a}{b}=\frac{a-c}{c-b}\)

Answer 1

Theo bài ta có:

\(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\)

\(\Rightarrow\frac{1}{c}=\frac{1}{2}\left(\frac{a}{ab}+\frac{b}{ab}\right)\)

\(\Rightarrow\frac{1}{c}=\frac{1}{2}.\frac{a+b}{ab}\)

\(\Rightarrow\frac{1}{c}=\frac{a+b}{2ab}\)

\(\Rightarrow2ab=c.\left(a+b\right)\)

\(\Rightarrow ab+ab=ca+cb\)

\(\Rightarrow ca-ab=ab-cb\)

\(\Rightarrow a\left(c-b\right)=b\left(a-c\right)\)

\(\Rightarrow\frac{c-b}{b}=\frac{a-c}{a}\Rightarrow\frac{a}{b}=\frac{a-c}{c-b}\left(đpcm\right)\)