Question

[Đinh Quốc Gia Nghĩa]

chứng minh a< thì a/b < a+1/b+1

Answer 1

[Capheny Bản Quyền]

\(\frac{a}{b}=\frac{a\left(b+1\right)}{b\left(b+1\right)}=\frac{ab+a}{b^2+1}\) 

\(\frac{a+1}{b+1}=\frac{b\left(a+1\right)}{b\left(b+1\right)}=\frac{ab+b}{b^2+1}\) 

Vì \(a< b\) 

nên \(\frac{ab+a}{b^2+1}< \frac{ab+b}{b^2+1}\) 

Vậy \(\frac{a}{b}< \frac{a+1}{b+1}\)

Answer 2

[Gukmin]

Ta có:\(\frac{a}{b}=\frac{a\left(b+1\right)}{b\left(b+1\right)}=\frac{ab+a}{b\left(b+1\right)}\)

\(\frac{a+1}{b+1}=\frac{b\left(a+1\right)}{b\left(b+1\right)}=\frac{ab+b}{b\left(b+1\right)}\)

Vì a < b => ab + a < ab + b

\(\Rightarrow\frac{ab+a}{b\left(b+1\right)}< \frac{ab+b}{b\left(b+1\right)}\)

\(\Rightarrow\frac{a}{b}< \frac{a+1}{b+1}\left(đpcm\right)\)

Vậy...

Linz