\(b^2=a\cdot c\)
=>\(\dfrac{b}{a}=\dfrac{c}{b}\)
\(c^2=b\cdot d\)
=>\(\dfrac{c}{b}=\dfrac{d}{c}\)
=>\(\dfrac{b}{a}=\dfrac{c}{b}=\dfrac{d}{c}\)
=>\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}\)
Đặt \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}=k\)
=>\(a=bk;b=ck;c=dk\)
\(b=c\cdot k=dk\cdot k=dk^2\)
\(a=bk=dk^2\cdot k=dk^3\)
\(\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}\)
\(=\dfrac{\left(dk^3\right)^3+\left(dk^2\right)^3+\left(dk\right)^3}{\left(dk^2\right)^3+\left(dk\right)^3+d^3}\)
\(=\dfrac{d^3k^9+d^3k^6+d^3k^3}{d^3k^6+d^3k^3+d^3}\)
\(=\dfrac{d^3\cdot k^3\left(k^6+k^3+1\right)}{d^3\left(k^6+k^3+1\right)}=k^3\)
\(\dfrac{a}{d}=\dfrac{dk^3}{d}=k^3\)
Do đó: \(\dfrac{a^3+b^3+c^3}{b^3+c^3+d^3}=\dfrac{a}{d}\)
