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a) \(b^2=ac\Rightarrow b.b=ac\Rightarrow\frac{b}{a}=\frac{c}{b}\Rightarrow\frac{a}{b}=\frac{b}{c}\)
\(c^2=bd\Rightarrow c.c=bd\Rightarrow\frac{c}{b}=\frac{d}{c}\Rightarrow\frac{b}{c}=\frac{c}{d}\)
Ngoặc ''}'' 2 điều trên
\(\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)
\(\Rightarrow\left(\frac{a}{b}\right)^3=\left(\frac{b}{c}\right)^3=\left(\frac{c}{d}\right)^3=\left(\frac{a+b-c}{b+c-d}\right)^3\)
\(\Rightarrow\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\left(\frac{a+b-c}{b+c-d}\right)^3\)
Vậy ...
b) \(b^2=ac\Rightarrow b.b=ac\Rightarrow\frac{b}{a}=\frac{c}{b}\Rightarrow\frac{a}{b}=\frac{b}{c}\)
\(c^2=bd\Rightarrow c.c=bd\Rightarrow\frac{c}{b}=\frac{d}{c}\Rightarrow\frac{b}{c}=\frac{c}{d}\)
Ngoặc ''}'' 2 điều trên
\(\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)
\(\Rightarrow\left(\frac{a}{b}\right)^3=\left(\frac{b}{c}\right)^3=\left(\frac{c}{d}\right)^3=\frac{a.b.c}{b.c.d}\)
\(\Rightarrow\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\frac{a}{d}\)(Do loại bỏ b.c trên tử + dưới mẫu nên còn a/d)
\(\Rightarrow\frac{a^3}{b^3}=\frac{8b^3}{3c^3}=\frac{125c^3}{125d^3}=\frac{a}{d}\)(Dùng tính chất phân số)
Vậy ...
P/s: Có gì khó hiểu thì hỏi nhé ^^
Giải:
Ta có: \(b^2=ac\Rightarrow\frac{a}{b}=\frac{b}{c}\)
\(c^2=bd\Rightarrow\frac{b}{c}=\frac{c}{d}\)
\(\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)
Lại có: \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\Rightarrow\frac{a^3}{b^3}=\frac{8b^3}{8c^3}=\frac{125c^3}{125d^3}=\frac{a^3+8b^3+125c^3}{b^3+8b^3+125c^3}\) (1)
Ta thấy \(\frac{a^3}{b^3}=\frac{a}{b}.\frac{b}{c}.\frac{c}{d}=\frac{a}{d}\) ( do \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\) ) (2)
Từ (1) và (2) \(\Rightarrow\frac{a}{d}=\frac{a^3+8b^3+125c^3}{b^3+8c^3+125d^3}\left(=\frac{a^3}{b^3}\right)\left(đpcm\right)\)
\(b^2=ac\Rightarrow\dfrac{a}{b}=\dfrac{b}{c};c^2=bd\Rightarrow\dfrac{b}{c}=\dfrac{c}{d}\\ \Rightarrow\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}\\ \Rightarrow\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{a^3+b^3+c^3}{c^3+b^3+d^3}\left(1\right)\\ \text{Đặt }\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}=k\\ \Rightarrow a=bk;b=ck;c=dk\\ \Rightarrow a=bk=ck^2=dk^3\\ \Rightarrow\dfrac{a}{d}=k^3\\ \text{Mà }\dfrac{a}{b}=k\Rightarrow\dfrac{a^3}{b^3}=k^3\\ \Rightarrow\dfrac{a}{d}=\dfrac{a^3}{b^3}\left(2\right)\\ \left(1\right)\left(2\right)\RightarrowĐpcm\)
Ta có :
\(\left\{{}\begin{matrix}b^2=ac\\c^2=bd\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=\dfrac{b^2}{c}\\d=\dfrac{c^2}{b}\end{matrix}\right.\)
\(\Rightarrow\dfrac{a}{d}=\dfrac{b^2}{c}:\dfrac{c^2}{b}\)
\(\Rightarrow\dfrac{a}{d}=\dfrac{b^2}{c}.\dfrac{b}{c^2}\)
\(\Rightarrow\dfrac{a}{d}=\dfrac{b^3}{c^3}=\dfrac{8b^3}{8c^3}=\dfrac{a^3}{b^3}=\dfrac{125c^3}{125d^3}\)
\(\Rightarrow\dfrac{a}{d}=\dfrac{a^3+8b^3+125c^3}{b^3+8c^3+125d^3}\left(dpcm\right)\)