Question

cho \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\). chứng minh rằng \(\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\frac{a}{d}\)

Answer 1

\(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)\(\Rightarrow\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\frac{abc}{bcd}\)\(=\frac{a^3+b^3+c^3}{b^3+c^3+d^3}\)

\(\Rightarrow\frac{a}{d}=\frac{a^3+b^3+c^3}{b^3+c^3+d^3}\)

=>Đpcm

Answer 2

đoàn cẩm lý sai rồi

Answer 3

Ta có: \(\frac{a}{b}=\frac{b}{c}\Rightarrow a=\frac{b^2}{c}\); \(\frac{b}{c}=\frac{c}{d}\Rightarrow d=\frac{c^2}{b}\)

Ta có vế trái  : \(\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\frac{\left(\frac{b^2}{c}\right)^3+b^3+c^3}{b^3+c^3+\left(\frac{c^2}{b}\right)^3}=\frac{\frac{b^6+b^3c^3+c^6}{c^3}}{\frac{b^6+b^3c^3+c^6}{b^3}}\)\(=\frac{b^6+b^3c^3+c^6}{c^3}\cdot\frac{b^3}{b^6+b^3c^3+c^6}=\frac{b^3}{c^3}\)

Ta có vế phải: \(\frac{a}{d}=\frac{\frac{b^2}{c}}{\frac{c^2}{b}}=\frac{b^2}{c}\cdot\frac{b}{c^2}=\frac{b^3}{c^3}\)

\(\Rightarrow\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\frac{a}{d}\)