Question

Cho a ∈ Z ; b ∈ N* ; n ∈ N*. Chứng minh rằng:

a) Nếu a < b thì \(\dfrac{a}{b}< \dfrac{a+n}{b+n}\)

b) Nếu a > b thì \(\dfrac{a}{b}>\dfrac{a+n}{b+n}\)

c) Nếu a = b thì \(\dfrac{a}{b}=\dfrac{a+n}{b+n}\)

Answer 1

a)
\(a< b\Leftrightarrow a.n< b.n\) ( Vì \(a\in Z;b\in\) N*\(;n\in\) N* )
\(\Leftrightarrow ab+a.n< ab+b.n\)
\(\Leftrightarrow a\left(b+n\right)< b\left(a+n\right)\)
\(\Leftrightarrow\dfrac{a}{b}< \dfrac{a+n}{b+n}\) ( đpcm )
Vậy........
b)
\(a>b\Leftrightarrow a.n>b.n\)
\(\Leftrightarrow ab+a.n>ab+b.n\)
\(\Leftrightarrow a\left(b+n\right)>b\left(a+n\right)\)
\(\Leftrightarrow\dfrac{a}{b}>\dfrac{a+n}{b+n}\) ( đpcm )
Vậy.........
c)
\(a=b\Leftrightarrow a.n=b.n\)
\(\Leftrightarrow ab+a.n=ab+b.n\)
\(\Leftrightarrow a\left(b+n\right)=b\left(a+n\right)\)
\(\Leftrightarrow\dfrac{a}{b}=\dfrac{a+n}{b+n}\) ( đpcm )
Vậy............