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Ta có :\(\frac{a}{b}=\frac{b}{c}\)
=> \(\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{a^2+b^2}{b^2+c^2}\)
=> \(\frac{a^2}{b^2}=\frac{a^2+b^2}{b^2+c^2}\)
=> \(\frac{a}{b}.\frac{a}{b}=\frac{a^2+b^2}{b^2+c^2}\)
=> \(\frac{a}{b}.\frac{b}{c}=\frac{a^2+b^2}{b^2+c^2}\)
=> \(\frac{a}{c}=\frac{a^2+b^2}{b^2+c^2}\left(\text{đpcm}\right)\)
\(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\)
=> \(\frac{2}{c}=\frac{1}{a}+\frac{1}{b}\)
=> \(\frac{2}{c}=\frac{a+b}{ab}\)
=> 2ab = ac + bc
=> ac + bc - 2ab = 0
=> (ac - ab) + (bc - ab) = 0
=> a(c - b) + b(c - a) = 0
=> a(c - b) = -b(c - a)
=> a(c - b) = b(a - c)
=> \(\frac{a}{b}=\frac{a-c}{c-b}\) (đpcm)
Từ \(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\Rightarrow\frac{1}{c}=\frac{1}{2}\left(\frac{a+b}{ab}\right)\)
\(\Rightarrow\frac{1}{c}=\frac{a+b}{2ab}\)
\(\Rightarrow2ab=c.\left(a+b\right)\)
\(\Rightarrow ab+ab=ac+bc\)
\(\Rightarrow ab-bc=ac-ab\)
\(\Rightarrow b.\left(a-c\right)=a.\left(c-b\right)\)
\(\Rightarrow\frac{a}{b}=\frac{a-c}{c-b}\)
ta có: \(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\)
\(=\frac{1}{c}\times2=\frac{1}{a}+\frac{1}{b}\)
\(=\frac{2}{c}=\frac{1}{a}+\frac{1}{b}\)
\(=\frac{2}{c}=\frac{b+a}{ab}\)
= \(c\left(b+a\right)=ab\times2\)
= cb +ca = ab+ab
= ab - cb = ac-ab
\(=b\left(a-c\right)=a\left(c-b\right)\)
= \(\frac{a}{b}=\frac{a-c}{c-b}\)
\(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\)
\(\frac{1}{c}=\frac{1}{2a}+\frac{1}{2b}\)
\(\frac{1}{c}=\frac{a+b}{2ab}\)
\(2ab=c\left(a+b\right)\)
\(ab+ab=ac+bc\)
\(ab-bc=ac-ab\)
\(b\left(a-c\right)=a\left(c-b\right)\)
\(\Rightarrow\frac{a}{b}=\frac{a-c}{c-b}\left(đpcm\right)\)
Ta có :
\(a^2=b.c\Rightarrow\frac{a}{c}=\frac{b}{a}=\frac{a+b}{c+a}=\frac{a-b}{c-a}\Rightarrow\frac{a+b}{a-b}=\frac{c+a}{c-a}\left(đpcm\right)\)
a) \(\frac{a}{a+b}=\frac{c}{c+d}\)=> a . ( c + d ) = c . ( a + b )
=> ac + ad = ac + cb
=> ad = bc
=> \(\frac{a}{b}=\frac{c}{d}\)
Từ a/b = b/c
Suy ra : bb = ac
b2 = ac
vậy : a2 + b2 / b2+ c2 = a2 + ac / ac + c2 = a(a+c) / c(a+c) = a/c
Vậy : Ta có được cái cần chứng minh :))
Lớp mình vừa kiểm tra 15' bài này xong .
cảm ơn bạn