ta có \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{d}=\dfrac{a+b+c}{b+c+d}\)
\(\Rightarrow\dfrac{a^3}{b^3}=\dfrac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}\)
\(\Leftrightarrow\dfrac{a}{b}\cdot\dfrac{b}{c}\cdot\dfrac{c}{d}=\dfrac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}\)
\(\Leftrightarrow\dfrac{a}{d}=\dfrac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}\left(đpcm\right)\)
Đặt \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\)
\(\Rightarrow a=bk;b=ck;c=dk\) (1)
Thay (1) vào đề bài:
\(VT=\left(\frac{bk+ck+dk}{ck+dk+d}\right)^3=\left[\frac{k\left(c+d\right)+bk}{k\left(c+d\right)+d}\right]^3=\left(\frac{bk}{d}\right)^3=\frac{bk}{d}\)
\(VP=\frac{bk}{d}\)
\(\Rightarrow VT=VP\)
hay \(\left(\frac{a+b+c}{b+c+d}\right)^3=\frac{a}{d}\rightarrowđpcm.\)