Question

Cho a,b,c \(\varepsilon\)R và a,b,c #0thõa mãn b²=ac.C/minh rằng \(\frac{a}{c}=\frac{\left(a+2007b\right)^2}{\left(b+2007c\right)^2}\)

Answer 1

Ta có:  \(b^2=a.c\Rightarrow\frac{a}{b}=\frac{b}{c}\)

Đặt \(\frac{a}{b}=\frac{b}{c}=k\left(k\in R\right)\)

\(\Rightarrow a=b.k\); \(b=c.k\)

\(\frac{a}{c}=\frac{a.c}{c.c}=\frac{b^2}{c^2}\left(1\right)\)

\(\frac{\left(a+2007b\right)^2}{\left(b+2007c\right)^2}=\frac{\left(b.k+2007b\right)^2}{\left(c.k+2007c\right)^2}=\frac{\left[b\left(k+2007\right)\right]^2}{\left[c.\left(k+2007\right)\right]^2}=\frac{b^2.\left(k+2007\right)^2}{c^2.\left(k+2007\right)^2}=\frac{b^2}{c^2}\left(2\right)\)

Từ \(\left(1\right)\) và \(\left(2\right)\) \(\Rightarrow\frac{a}{c}=\frac{\left(a+2007b\right)^2}{\left(b+2007c\right)^2}\) \(\left(đpcm\right)\)