Question

cho  a , b , c , d khác 0

b²_=a.c

_c²=b.d

_{cmr: (a^3 + b^3 + c^3) / (b^3+c^3+d^3) = [(a+b+c)^3] / [(b+c+d)^3]}

các biểu thức trên có nghĩa

Answer 1

Ta có: \(\hept{\begin{cases}b^2=ac\\c^2=bd\end{cases}\Rightarrow}\hept{\begin{cases}\frac{a}{b}=\frac{b}{c}\\\frac{b}{c}=\frac{c}{d}\end{cases}}\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)

Đặt: \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=t\)

\(\Rightarrow\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=t^3\Rightarrow\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=t^3\)(1)

\(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=\frac{a+b+c}{b+c+d}=t\Rightarrow\frac{\left(a+b+c\right)^3}{\left(b+c+d\right)^3}=t^3\)(2)

(1); (2) => đpcm