Giải pt :\(\frac{\sqrt{x-2009}-1}{x-2009}+\frac{\sqrt{y-2010}-1}{y-2010}+\frac{\sqrt{z-2011}-1}{z-2011}=\frac{3}{4}\)
\(s=\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+...+\frac{1}{\sqrt{2009}+\sqrt{2011}}\)
Chứng minh : \(\frac{1}{\left(\sqrt{2}+\sqrt{5}\right)^3}+\frac{1}{\left(\sqrt{5}+\sqrt{8}\right)^3}\)\(+...+\frac{1}{\left(\sqrt{2006}+\sqrt{2009}\right)^3}\)\(< \frac{11}{135}\)
giải phương trình:\(\sqrt{x-2}+\sqrt{y+2009}+\sqrt{z-2010}=\frac{1}{2}\left(x+y+z\right)\)
1-a,\(A=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{n-1}+\sqrt{n}}\)
b,\(B=\frac{1}{2+\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}\)
tính:\(\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-\frac{1}{\sqrt{4}-\sqrt{5}}+\frac{1}{\sqrt{5}-\sqrt{6}}-\frac{1}{\sqrt{6}-\sqrt{7}}+\frac{1}{\sqrt{7}-\sqrt{8}}-\frac{1}{\sqrt{8}-\sqrt{9}}\)
Tính :
a ) \(S=\frac{1}{\sqrt{1}\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{7}}+.....+\)\(\frac{1}{\sqrt{2017}+\sqrt{2019}}\)
b ) \(S=\frac{1}{\sqrt{2}+\sqrt{4}}+\frac{1}{\sqrt{4}+\sqrt{6}}+....+\frac{1}{\sqrt{100}+\sqrt{102}}\)
c ) \(S=\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+.....+\frac{1}{\sqrt{100}+\sqrt{101}}\)
d ) \(S=\frac{1}{\sqrt{3}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{9}}+\frac{1}{\sqrt{9}+\sqrt{12}}+....+\frac{1}{\sqrt{2016}+\sqrt{2019}}\)
Rút gọn:
\(A=\frac{1}{1-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}+...+\frac{1}{\sqrt{99}-\sqrt{100}}\)
Rút gọn các biểu thức,
a> A= \(\frac{1}{\sqrt{1}+\sqrt{2}}\) + \(\frac{1}{\sqrt{2}+\sqrt{3}}\)+ \(\frac{1}{\sqrt{3}+\sqrt{4}}\)+ ......... + \(\frac{1}{\sqrt{n-1}+\sqrt{n}}\)
b> B= \(\frac{1}{\sqrt{1}-\sqrt{2}}\)- \(\frac{1}{\sqrt{2}-\sqrt{3}}\)- \(\frac{1}{\sqrt{3}-\sqrt{4}}\)- .......... - \(\frac{1}{\sqrt{24}-\sqrt{25}}\)