Question

So sánh : a) \(\frac{1}{n}\) và \(\frac{1}{n+1}\) ; b) \(\frac{n+1}{n+2}\) và \(\frac{n}{n+3}\) (với n \(\in\) N*)

Answer 1

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

\(\frac{1}{n+1}=\frac{n}{n\left(n+1\right)}\)

n+1>n

Do đó: \(\frac{1}{n}>\frac{1}{n+1}\)

b) Ta có: \(\frac{n+1}{n+2}=\frac{\left(n+1\right)\left(n+3\right)}{\left(n+2\right)\left(n+3\right)}=\frac{n^2+4n+3}{\left(n+2\right)\left(n+3\right)}\)

\(\frac{n}{n+3}=\frac{n\left(n+2\right)}{\left(n+2\right)\left(n+3\right)}=\frac{n^2+2n}{\left(n+2\right)\left(n+3\right)}\)

\(n^2+4n+3>n^2+2n\)

Do đó: \(\frac{n+1}{n+2}>\frac{n}{n+3}\)