Ta có:
\(b^2=c\cdot a\Rightarrow\dfrac{a}{b}=\dfrac{b}{c}\)
\(c^2=b\cdot d\Rightarrow\dfrac{b}{c}=\dfrac{c}{d}\)
Suy ra: \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}\)
\(\Rightarrow\left(\dfrac{a}{b}\right)^3=\left(\dfrac{b}{c}\right)^3=\left(\dfrac{c}{d}\right)^3=\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}\)
Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:
\(\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}\)(1)
Mặt khác: \(\dfrac{a^3}{b^3}=\left(\dfrac{a}{b}\right)^3=\dfrac{a}{b}\cdot\dfrac{b}{c}\cdot\dfrac{c}{d}=\dfrac{a}{d}\)(2)
Từ (1) và (2) \(\Rightarrow\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}=\dfrac{a}{d}\)
\(Toru\)
\(b^2=ac,c^2=bd\Rightarrow\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}\)
=>\(\dfrac{a^3}{b^3}=\dfrac{b^3}{c^3}=\dfrac{c^3}{d^3}=\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}\)(1)
ma \(\dfrac{a^3}{b^3}=\dfrac{a.b.c}{b.c.d}=\dfrac{a}{d}\)
=> dpcm