Question

So sánh: n + 1/ n + 2 và n/ n + 3

 

Answer 1

\(\frac{n+1}{n+2}\)và \(\frac{n}{n+3}\)
\(\orbr{\begin{cases}\frac{n+1}{n+2}=\frac{\left(n+1\right)\left(n+3\right)}{\left(n+2\right)\left(n+3\right)}=\frac{n^2+3n+n+3}{\left(n+2\right)\left(n+3\right)}=\frac{n^2+4n+3}{\left(n+2\right)\left(n+3\right)}\\\frac{n}{n+3}=\frac{n\left(n+2\right)}{\left(n+2\right)\left(n+3\right)}=\frac{n^2+2n}{\left(n+2\right)\left(n+3\right)}\end{cases}}\)\(\Rightarrow\frac{n^2+4n+3}{\left(n+2\right)\left(n+3\right)}>\frac{n^2+2n}{\left(n+2\right)\left(n+3\right)}\Rightarrow\frac{n+1}{n+2}>\frac{n}{n+3}\)

Answer 2

Ta nhân chéo (n+1) x (n+3)=n^2+n+3n+3 (1)

n x (n+2)=n^2+2n (2)

Ta thấy (1)>(2) do n^2+n+3n+3 >  n^2+2n  nên (n+1) x (n+3) > n x (n+2)

Từ đó suy ra n+1/n+2 > n/n+3 ( tính chất )