CMR: \(\frac{1}{\sqrt{1}+\sqrt{2}}\) = \(\sqrt{2}\)- 1
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T
27 tháng 6 2019
Xét số hạng tổng quát: \(\frac{1}{\sqrt{n}+\sqrt{n+1}}=\frac{\sqrt{n+1}-\sqrt{n}}{\left(\sqrt{n+1}+\sqrt{n}\right)\left(\sqrt{n+1}-\sqrt{n}\right)}=\sqrt{n+1}-\sqrt{n}\) (do \(\sqrt{n+1}-\sqrt{n}>0\forall n\in\mathbb{N}\text{ nên ta có thể nhân liên hợp}\))
Áp dụng vào và ta có:
\(VT=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+...+\sqrt{2013^2}-\sqrt{2013^2-1}\)
\(=\sqrt{2013^2}-1=2013-1=2012^{\left(đpcm\right)}\)
ML
12 tháng 8 2015
\(\frac{\sqrt{1+\frac{2\sqrt{2}}{3}}+\sqrt{1-\frac{2\sqrt{2}}{3}}}{\sqrt{1+\frac{2\sqrt{2}}{3}}-\sqrt{1-\frac{2\sqrt{2}}{3}}}=\frac{\sqrt{\frac{3+2\sqrt{2}}{3}}+\sqrt{\frac{3-2\sqrt{2}}{3}}}{\sqrt{\frac{3+2\sqrt{2}}{3}}-\sqrt{\frac{3-2\sqrt{2}}{3}}}=\frac{\frac{\sqrt{\left(1+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{2}-1\right)^2}}{\sqrt{3}}}{\frac{\sqrt{\left(1+\sqrt{2}\right)^2}-\sqrt{\left(\sqrt{2}-1\right)^2}}{\sqrt{3}}}\)\(=\frac{1+\sqrt{2}+\sqrt{2}-1}{1+\sqrt{2}-\sqrt{2}+1}=\frac{2\sqrt{2}}{2}=\sqrt{2}\left(đpcm\right)\)
HL
8 tháng 7 2019
Ta có \(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{47}+\sqrt{48}}=\frac{1-\sqrt{2}}{\left(1-\sqrt{2}\right)\left(1+\sqrt{2}\right)}+\frac{\sqrt{2}-\sqrt{3}}{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}+\frac{\sqrt{3}-\sqrt{4}}{\left(\sqrt{3}-\sqrt{4}\right)\left(\sqrt{3}+\sqrt{4}\right)}\)
B
8 tháng 7 2019
Ta có:
\(\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{47}+\sqrt{48}}\)
\(=\frac{\sqrt{1}-\sqrt{2}}{-1}+\frac{\sqrt{2}-\sqrt{3}}{-1}+\frac{\sqrt{3}-\sqrt{4}}{-1}+...+\frac{\sqrt{47}-\sqrt{48}}{-1}\)
\(=\frac{\sqrt{1}-\sqrt{2}+\sqrt{2}-\sqrt{3}+\sqrt{3}-\sqrt{4}+...+\sqrt{47}-\sqrt{48}}{-1}\)
\(=\frac{\sqrt{1}-\sqrt{48}}{-1}\)
\(=4\sqrt{3}-1\approx5,9>3\left(đpcm\right)\)
B
28 tháng 7 2019
\(a,A=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+..+\frac{1}{\sqrt{99}+\sqrt{100}}\)
\(=\frac{\sqrt{1}-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+...+\frac{\sqrt{99}-\sqrt{100}}{99-100}\)
\(=\frac{1-\sqrt{2}+\sqrt{2}-\sqrt{3}+...+\sqrt{99}-\sqrt{100}}{-1}\)
\(=\frac{1-\sqrt{100}}{-1}=9\)
\(b,B=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+..+\frac{1}{\sqrt{99}}\)
\(=\frac{2}{\sqrt{1}+\sqrt{1}}+\frac{2}{\sqrt{2}+\sqrt{2}}+\frac{2}{\sqrt{3}+\sqrt{3}}+...+\frac{2}{\sqrt{99}+\sqrt{99}}>\frac{2}{\sqrt{1}+\sqrt{2}}+\frac{2}{\sqrt{2}+\sqrt{3}}+...+\frac{2}{\sqrt{99}+\sqrt{100}}\)\(\Rightarrow B>2\left(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+..+\frac{1}{\sqrt{99}+\sqrt{100}}\right)\)
\(\Rightarrow B>2\left(\frac{\sqrt{1}-\sqrt{2}+\sqrt{2}-\sqrt{3}+...+\sqrt{99}-\sqrt{100}}{-1}\right)\)
\(\Rightarrow B>2\left(\frac{1-\sqrt{100}}{-1}\right)\)
\(\Rightarrow B>2.9=18\left(ĐPCM\right)\)
1 tháng 10 2016
Xét với n là số tự nhiên không nhỏ hơn 1
Ta có : \(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n\left(n+1\right)}\left(\sqrt{n}+\sqrt{n+1}\right)}=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n}.\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
Áp dụng điều trên ta có
\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{2002\sqrt{2001}+2001\sqrt{2002}}\)
\(=1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2001}}-\frac{1}{\sqrt{2002}}\)
\(=1-\frac{1}{\sqrt{2002}}< 1-\frac{1}{\sqrt{2025}}=1-\frac{1}{45}=\frac{44}{45}\)
1 tháng 10 2016
ta chứng minh công thức tổng quát sau
\(\frac{1}{\left[n+1\right]\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n\left[n+1\right]}\left[\sqrt{n+1}+\sqrt{n}\right]}\)
=\(\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left[n+1\right]}\left[n+1-n\right]}=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left[n+1\right]}}\)
=\(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
ta có \(\frac{1}{2\sqrt{1}+1\sqrt{2}}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}\)
\(\frac{1}{3\sqrt{2}+2\sqrt{3}}=\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\)
........
\(\frac{1}{2002\sqrt{2001}+2001\sqrt{2002}}=\frac{1}{\sqrt{2001}}-\frac{1}{\sqrt{2002}}\)
=> \(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+..+\frac{1}{2002\sqrt{2001}+2001\sqrt{2002}}\)
=\(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2001}}-\frac{1}{\sqrt{2002}}\)
=\(1-\frac{1}{\sqrt{2002}}< \frac{44}{45}\)
Xét :\(\frac{1}{\sqrt{1}+\sqrt{2}}=\frac{1}{\sqrt{2}-1}\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}\right)\)
\(=\frac{1}{\sqrt{2}-1}\left(\frac{\sqrt{2}-1}{\sqrt{1}.\sqrt{2}}\right)\)
\(=\sqrt{2}-1\)
Nhân cả tử cả mẫu cho \(\sqrt{2}-1\)
\(\frac{\sqrt{2}-1}{\left(\sqrt{2}-1\right)\left(\sqrt{1}+\sqrt{2}\right)}=\sqrt{2}-1\)
\(\frac{\sqrt{2}-1}{1}=\sqrt{2}-1\)vì \(\left(\sqrt{2}-1\right)\left(\sqrt{1}+\sqrt{2}\right)=1\)