Question

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

Answer 1

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

áp dụng t/c dãy tỉ số bằng nhau ta có:

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

\(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\Rightarrow\frac{a^3}{b^3}=\frac{a}{b}\cdot\frac{a}{b}\cdot\frac{a}{b}=\frac{a}{b}\cdot\frac{c}{d}\cdot\frac{b}{c}=\frac{a}{d}\left(2\right)\)

từ (1) và (2) => \(\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\left(\frac{a}{d}\right)^3\left(đpcm\right)\)