n+1/n+2<1
Suy ra n+1/n+2<n+2/n+1+2=n+2/n+3

\(\dfrac{n+1}{2n+3}\) < \(\dfrac{n+1}{2n+2}\) < \(\dfrac{n+2}{2n+2}\)

Ta có : \(\frac{n+1}{n+5}< \frac{n+1}{n+3}\)   mà \(\frac{n+1}{n+3}< \frac{n+2}{n+3}\)

\(\Rightarrow\)\(\frac{n+1}{n+5}< \frac{n+2}{n+3}\)\(,\forall\)\(n\in N\)

Đặt : \(A=\dfrac{n+1}{n+5}\) và \(B=\dfrac{n+3}{n+4}\).

Ta có : \(A=\dfrac{n+1}{n+5}=\dfrac{n+5-4}{n+5}=\dfrac{n+5}{n+5}-\dfrac{4}{n+5}=1-\dfrac{4}{n+5}\)

Và : \(B=\dfrac{n+3}{n+4}=\dfrac{n+4-1}{n+4}=\dfrac{n+4}{n+4}-\dfrac{1}{n+4}=1-\dfrac{1}{n+4}\)

Cả \(A\) và \(B\) đều có hạng tử \(1\) nên ta so sánh : \(\dfrac{4}{n+5}\) và \(\dfrac{1}{n+4}\).

Quy đồng ta được : 

\(\dfrac{4\left(n+4\right)}{\left(n+5\right)\left(n+4\right)}=\dfrac{4n+16}{\left(n+5\right)\left(n+4\right)}\) và \(\dfrac{n+5}{\left(n+4\right)\left(n+5\right)}\).

Do mẫu bằng nhau nên ta so sánh tử, ta thấy : 

\(4n+16-\left(n+5\right)=4n+16-n-5=3n+11\).

Do \(n\) là số tự nhiên nên \(3n\ge0\), suy ra \(3n+11\ge11\).

Suy ra được : \(4n+16-\left(n+5\right)=3n+11\ge11>0\) nên \(4n+16>n+5\).

Do đó, \(\dfrac{4}{n+5}>\dfrac{4}{n+4}\Rightarrow1-\dfrac{4}{n+5}< 1-\dfrac{4}{n+4}\).

Vậy : \(A< B\) hay \(\dfrac{n+1}{n+5}< \dfrac{n+3}{n+4}\).

Lời giải:

$\frac{n+3}{n+4}=\frac{(n+4)-1}{n+4}=1-\frac{1}{n+4}$

$\frac{n+1}{n+2}=\frac{(n+2)-1}{n+2}=1-\frac{1}{n+2}$

Vì $n+4> n+2$ nên $\frac{1}{n+4}< \frac{1}{n+2}$

Suy ra $1-\frac{1}{n+4}> 1-\frac{1}{n+2}$

Hay $\frac{n+3}{n+4}> \frac{n+1}{n+2}$

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$\frac{n-1}{n+4}< \frac{n-1}{n+2}=\frac{(n+2)-3}{n+2}=1-\frac{3}{n+2}$

$<1-\frac{n+3}=\frac{n}{n+3}$