Question

Cho \(\dfrac{a}{c}=\dfrac{c}{b}=\dfrac{b}{d}\)

CMR:\(\dfrac{a^3+c^3-b^3}{c^3+b^3-d^3}=\dfrac{a}{d}\)

Answer 1

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

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

Từ \(\dfrac{a}{c}=\dfrac{c}{b}=\dfrac{b}{d}\)

Ta xét tích: \(\left(\dfrac{a}{c}\right)^3=\dfrac{a}{c}.\dfrac{a}{c}.\dfrac{a}{c}=\dfrac{a}{c}.\dfrac{c}{b}.\dfrac{b}{d}=\dfrac{a}{d}\left(2\right)\)

Từ (1) và (2) \(\Rightarrow\dfrac{a^3+c^3-b^3}{c^3+b^3-d^3}=\dfrac{a}{d}\left(dpcm\right)\)