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Với mọi n nguyên dương ta có:
\(\left(\sqrt{n+1}+\sqrt{n}\right)\left(\sqrt{n+1}-\sqrt{n}\right)=1\Rightarrow\frac{1}{\sqrt{n+1}+\sqrt{n}}=\sqrt{n+1}-\sqrt{n}\)
Với k nguyên dương thì
\(\frac{1}{\sqrt{k-1}+\sqrt{k}}>\frac{1}{\sqrt{k+1}+\sqrt{k}}\Rightarrow\frac{2}{\sqrt{k-1}+\sqrt{k}}>\frac{1}{\sqrt{k-1}+\sqrt{k}}+\frac{1}{\sqrt{k+1}+\sqrt{k}}=\sqrt{k}-\sqrt{k-1}+\sqrt{k+1}-\sqrt{k}\)
\(=\sqrt{k+1}-\sqrt{k-1}\)(*)
Đặt A = vế trái. Áp dụng (*) ta có:
\(\frac{2}{\sqrt{1}+\sqrt{2}}>\sqrt{3}-\sqrt{1}\)
\(\frac{2}{\sqrt{3}+\sqrt{4}}>\sqrt{5}-\sqrt{3}\)
...
\(\frac{2}{\sqrt{79}+\sqrt{80}}>\sqrt{81}-\sqrt{79}\)
Cộng tất cả lại
\(2A=\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{3}+\sqrt{4}}+....+\frac{2}{\sqrt{79}+\sqrt{80}}>\sqrt{81}-1=8\Rightarrow A>4\left(đpcm\right)\)
3.
Theo bất đẳng thức cô si ta có:
\(\sqrt{b-1}=\sqrt{1.\left(b-1\right)}\le\frac{1+b-1}{2}=\frac{b}{2}\Rightarrow a.\sqrt{b-1}\le\frac{a.b}{2}\)
Tương tự \(\Rightarrow b.\sqrt{a-1}\le\frac{a.b}{2}\Rightarrow a.\sqrt{b-1}+b.\sqrt{a-1}\le a.b\)
Dấu "=" xảy ra khi và chỉ khi \(a=b=2\)
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)
\(\frac{1}{1+\sqrt{2}}=\frac{1}{2\sqrt{1}+2\sqrt{2}}+\frac{1}{2\sqrt{1}+2\sqrt{2}}>\frac{1}{2\sqrt{1}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+2\sqrt{3}}\)
\(\Rightarrow\frac{1}{\sqrt{1}+\sqrt{2}}>\frac{1}{2}\left(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}\right)=\frac{1}{2}\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}\right)\)
Tương tự với các biểu thức còn lại rồi cộng vế với vế ta có:
\(VT>\frac{1}{2}\left(\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{81}-\sqrt{80}\right)=\frac{1}{2}\left(\sqrt{81}-1\right)=4\)
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)
\(\frac{1}{\sqrt{1}+\sqrt{2}}+....\frac{1}{\sqrt{79}+\sqrt{80}}>\frac{1}{\sqrt{100}}+...+\frac{1}{\sqrt{100}}\) (40 số)
................................................................\(>\frac{40}{10}=4\)
=>đpcm
hc tốt
ko chắc lắm :)
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 )
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à ừ cảm ơn bạn nhìu nha