Question

Cho a,b\(\in Z;a< b;b>0\).Chứng minh rằng \(\dfrac{a}{b}< \dfrac{a+2006}{b+2006}\)

Answer 1

Ta có:

\(\dfrac{a}{b}=\dfrac{a\cdot\left(b+2006\right)}{b\cdot\left(b+2006\right)}=\dfrac{ab+2006a}{b^2+2006b}\\ \dfrac{a+2006}{b+2006}=\dfrac{\left(a+2006\right)\cdot b}{\left(b+2006\right)\cdot b}=\dfrac{ab+2006b}{b^2+2006b}\)

Vì \(a< b\Rightarrow2006a< 2006b\Rightarrow ab+2006a< ab+2006b\)

\(\Rightarrow\dfrac{ab+2006a}{b^2+2006b}< \dfrac{ab+2006b}{b^2+2006b}\\ \Leftrightarrow\dfrac{a}{b}< \dfrac{a+2006}{b+2006}\)