\(\frac{a}{c}=\frac{c}{b}\Rightarrow ab=c^2\)
a, \(\frac{a^2+c^2}{b^2+c^2}=\frac{a^2+ab}{b^2+ab}=\frac{a\left(a+b\right)}{b\left(a+b\right)}=\frac{a}{b}\)
b, \(\frac{b^2-a^2}{a^2+c^2}=\frac{\left(b-a\right)\left(b+a\right)}{a^2+ab}=\frac{\left(b-a\right)\left(a+b\right)}{a\left(a+b\right)}=\frac{b-a}{a}\)
Mình gọi a1,a2,a3,a4 là a,b,c,d nha
Ta có: \(\hept{\begin{cases}b^2=a\cdot c\\c^2=b\cdot d\end{cases}\Rightarrow\hept{\begin{cases}\frac{a}{b}=\frac{b}{c}\\\frac{b}{c}=\frac{c}{d}\end{cases}}}\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)
\(\Rightarrow\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\frac{a\cdot b\cdot c}{b\cdot c\cdot d}=\frac{a}{d}\)\(\left(1\right)\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta được:
\(\Rightarrow\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\frac{a^3+b^3+c^3}{b^3+c^3+d^3}\)\(\left(2\right)\)
Từ (1) và (2)
\(\Rightarrow\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\frac{a}{d}\)(\(\left(ĐPCM\right)\)
