Question

Cho \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)CMR:\(\left(\frac{a+b+c}{b+c+d}\right)^3=\frac{a}{d}\)

Answer 1

Đặt \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\)

\(\Rightarrow k=\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=\frac{a+b+c}{b+c+d}\Rightarrow k^3=\left(\frac{a+b+c}{b+c+d}\right)^3\left(1\right)\)

Mà \(k^3=\frac{a}{b}.\frac{b}{c}.\frac{c}{d}=\frac{a}{d}\left(2\right)\)

Từ (1),(2)\(\Rightarrow\left(\frac{a+b+c}{b+c+d}\right)^3=\frac{a}{d}\left(đpcm\right)\)

Answer 2

Ta có:\(\frac{a}{b}\)=\(\frac{b}{c}\)=\(\frac{c}{d}\)

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

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

\(\Rightarrow\)đpcm