Ta có:

\(\frac{1}{\sqrt{1}+\sqrt{2}}>\frac{1}{\sqrt{2}+\sqrt{3}};\frac{1}{\sqrt{3}+\sqrt{4}}>\frac{1}{\sqrt{4}+\sqrt{5}};...;\frac{1}{\sqrt{79}+\sqrt{80}}>\frac{1}{\sqrt{80}+\sqrt{81}}\)

Do đó \(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}\)

\(=\frac{1}{2}\left(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{79}+\sqrt{80}}\right)\)\(>\frac{1}{2}\left(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{80}+\sqrt{81}}\right)\)

\(=\frac{1}{2}\left(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{80}-\sqrt{79}+\sqrt{81}-\sqrt{80}\right)\)

\(=\frac{1}{2}\left(-\sqrt{1}+\sqrt{81}\right)=\frac{1}{2}\left(-1+9\right)=4\)

Suy ra đpcm.

Đặt \(A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{80}+\sqrt{79}}\)
Suy ra 
\(2A=2\left(\frac{1}{\sqrt{2}+\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}\right)\)
\(=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{3}+\sqrt{4}}...+\frac{1}{\sqrt{79}+\sqrt{80}}\)
\(>\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{80}+\sqrt{81}}\)
\(=\left(\sqrt{2}-\sqrt{1}\right)+\left(\sqrt{3}-\sqrt{2}\right)+....+\left(\sqrt{80}-\sqrt{79}\right)+\left(\sqrt{81}-\sqrt{79}\right)\)
\(=\sqrt{81}-1=9-1=8\Rightarrow2A>8\Leftrightarrow A>8\)( Đpcm)

Đặt \(A=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}\)

Ta có: \(\frac{1}{1+\sqrt{2}}>\frac{1}{2}\left(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}\right)\)

\(\frac{1}{\sqrt{3}+\sqrt{4}}>\frac{1}{2}\left(\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}\right)\)

...

\(\frac{1}{\sqrt{79}+\sqrt{80}}>\frac{1}{2}\left(\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{80}+\sqrt{81}}\right)\)

Cộng các bất đẳng thức trên lại với nhau, ta được:

\(A>\frac{1}{2}\left(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{80}+\sqrt{81}}\right)\)

\(\Leftrightarrow A>\frac{1}{2}\left(\frac{\sqrt{2}-1}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+...+\frac{\sqrt{81}-\sqrt{80}}{81-80}\right)\)

\(\Leftrightarrow A>\frac{1}{2}\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{81}-\sqrt{80}\right)\)

\(\Leftrightarrow A>\frac{1}{2}\left(\sqrt{81}-1\right)=\frac{1}{2}\cdot\left(9-1\right)=\frac{1}{2}\cdot8=4\)

\(\Leftrightarrow A>4\)(đpcm)

Tổng quát ta có: Với \(n\inℕ\)ta có:

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

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

Với \(n=2\)\(\Rightarrow\frac{1}{\sqrt{2}+\sqrt{3}}=\sqrt{3}-\sqrt{2}\)

Với \(n=3\)\(\Rightarrow\frac{1}{\sqrt{3}+\sqrt{4}}=\sqrt{4}-\sqrt{3}\)

...........................

Với \(n=79\)\(\Rightarrow\frac{1}{\sqrt{79}+\sqrt{80}}=\sqrt{80}-\sqrt{79}\)

\(\Rightarrow\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+.....+\frac{1}{\sqrt{79}+\sqrt{80}}\)

\(=\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+......+\sqrt{80}-\sqrt{79}\)

\(=\sqrt{80}-\sqrt{2}=\sqrt{40.2}-\sqrt{2}=\sqrt{40}.\sqrt{2}-\sqrt{2}\)

\(=\sqrt{2}.\left(\sqrt{40}-1\right)>\sqrt{2}.\left(\sqrt{36}-1\right)\)

\(=\sqrt{2}.\left(6-1\right)=5\sqrt{2}>4\)( đpcm )

Ta chứng minh được \(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\sqrt{n+1}-\sqrt{n}\) với mọi n là số tự nhiên lớn hơn 0

Đặt \(A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}\)

Ta có \(2A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{79}+\sqrt{80}}>\)

\(>\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{5}}+...+\frac{1}{\sqrt{79}+\sqrt{80}}+\frac{1}{\sqrt{80}+\sqrt{81}}\)

\(=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+\sqrt{5}-\sqrt{4}+...+\sqrt{80}-\sqrt{79}+\sqrt{81}-\sqrt{80}\)

\(=\sqrt{81}-\sqrt{1}=8\)

\(\Rightarrow2A>8\Rightarrow A>4\)

\(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\frac{\sqrt{n+1}-\sqrt{n}}{n+1-n}=\sqrt{n+1}-\sqrt{n}\)

\(\rightarrow\)VT = \(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{80}-\sqrt{79}=\sqrt{80}-1>4\)

Sai đề nha bạn, 2 số dưới mẫu cuối cùng là \(\sqrt{79}\) và \(\sqrt{80}\) mới theo quy luật 

Nhận xét: với mọi \(a\inℕ^∗\) ta có : 

\(\frac{1}{\sqrt{a-1}+\sqrt{a}}>\frac{1}{\sqrt{a+1}+\sqrt{a}}\)\(\Leftrightarrow\)\(\frac{2}{\sqrt{a-1}+\sqrt{a}}=\frac{1}{\sqrt{a-1}+\sqrt{a}}+\frac{1}{\sqrt{a-1}+\sqrt{a}}>\frac{1}{\sqrt{a-1}+\sqrt{a}}+\frac{1}{\sqrt{a+1}+\sqrt{a}}\)

\(=\frac{\sqrt{a}-\sqrt{a-1}}{\left(\sqrt{a-1}+\sqrt{a}\right)\left(\sqrt{a}-\sqrt{a-1}\right)}+\frac{\sqrt{a+1}-\sqrt{a}}{\left(\sqrt{a+1}+\sqrt{a}\right)\left(\sqrt{a+1}-\sqrt{a}\right)}\)

\(=\sqrt{a}-\sqrt{a-1}+\sqrt{a+1}-\sqrt{a}=\sqrt{a+1}-\sqrt{a-1}\)

\(\Rightarrow\)\(2B=\frac{2}{1+\sqrt{2}}+\frac{2}{\sqrt{3}+\sqrt{4}}+\frac{2}{\sqrt{5}+\sqrt{6}}+...+\frac{2}{\sqrt{79}+\sqrt{80}}\)

\(>\sqrt{3}-1+\sqrt{5}-\sqrt{3}+\sqrt{7}-\sqrt{5}+...+\sqrt{81}-\sqrt{79}\)

\(=\sqrt{81}-1=9-1=8\)

\(2B>8\)\(\Rightarrow\)\(B>\frac{8}{2}=4\) ( đpcm ) 

... 

à ừ cảm ơn bạn nhìu nha

\(\frac{1}{1+\sqrt{2}}=\frac{1}{2\sqrt{1}+2\sqrt{2}}+\frac{1}{2\sqrt{1}+2\sqrt{2}}>\frac{1}{2\sqrt{1}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+2\sqrt{3}}\)

\(\Rightarrow\frac{1}{\sqrt{1}+\sqrt{2}}>\frac{1}{2}\left(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}\right)=\frac{1}{2}\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}\right)\)

Tương tự với các biểu thức còn lại rồi cộng vế với vế ta có:

\(VT>\frac{1}{2}\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{81}-\sqrt{80}\right)=\frac{1}{2}\left(\sqrt{81}-1\right)=4\)