Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}=\dfrac{a+b-c}{b+c-d}\Rightarrow\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{\left(a+b-c\right)^3}{\left(b+c-d\right)^3}=\left(\dfrac{a+b-c}{b+c-d}\right)^3\left(1\right)\)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}\left(2\right)\)
Từ \(\left(1\right),\left(2\right)\Rightarrow\dfrac{a}{b}=\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}=\left(\dfrac{a+b-c}{b+c-d}\right)^3\)
Vậy \(\dfrac{a}{b}=\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}=\left(\dfrac{a+b-c}{b+c-d}\right)^3\left(dpcm\right)\)
