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Ta có: \(A=\sqrt{2012}-\sqrt{2011}=\frac{1}{\sqrt{2012}+\sqrt{2011}}< \frac{1}{\sqrt{2011}+\sqrt{2010}}\)
\(=\sqrt{2011}-\sqrt{2010}< \sqrt{2011}.\sqrt{2010}=B\)
Vậy A<B
\(C=\sqrt[3]{2011}-\sqrt[3]{2010}=\frac{2011-2010}{\left(\sqrt[3]{2011^2}+\sqrt[3]{2011}\sqrt[3]{2010}+\sqrt[3]{2010^2}\right)}=\frac{1}{\left(\sqrt[3]{2011^2}+\sqrt[3]{2011}\sqrt[3]{2010}+\sqrt[3]{2010^2}\right)}\)
\(B=\sqrt[3]{2010}-\sqrt[3]{2009}=\frac{2010-2009}{\left(\sqrt[3]{2010^2}+\sqrt[3]{2010}\sqrt[3]{2009}+\sqrt[3]{2009^2}\right)}=\frac{1}{\left(\sqrt[3]{2010^2}+\sqrt[3]{2010}\sqrt[3]{2009}+\sqrt[3]{2009^2}\right)}\)Vì \(\left(\sqrt[3]{2011^2}+\sqrt[3]{2011}\sqrt[3]{2010}+\sqrt[3]{2010^2}\right)>\left(\sqrt[3]{2010^2}+\sqrt[3]{2010}\sqrt[3]{2009}+\sqrt[3]{2009^2}\right)\)
\(B< C\)
\(\left(\sqrt{2015}+\sqrt{2018}\right)^2=4033+2\sqrt{2015\cdot2018}\)
\(\left(\sqrt{2016}+\sqrt{2017}\right)^2=4033+2\sqrt{2016\cdot2017}\)
\(2015\cdot2018=2015\cdot2017+2015=2017\cdot\left(2015+1\right)-2017+2015\)
\(=2017\cdot2016-2\)
\(\Rightarrow2015\cdot2018< 2016\cdot2017\)
\(\Rightarrow\sqrt{2015}+\sqrt{2018}< \sqrt{2016}+\sqrt{2017}\)
có bạn nào giải thích cho mình từ đoạn 2015.2018=2015.2017+2015 trở đi được k? mình cảm ơn
Ta gán : \(1992\rightarrow D\); \(1992\rightarrow A\)
\(D=D+1:A=D.\sqrt[D]{A}\)
CALC , bấm liên tiếp dấu "=" cho đến khi D = 2013 thì dừng.
Sau đó bấm \(\frac{Ans}{D}\) sẽ ra kết quả cần tính.
+ \(\sqrt{2013}-\sqrt{2011}=\frac{\left(\sqrt{2013}-\sqrt{2011}\right)\left(\sqrt{2013}+\sqrt{2011}\right)}{\sqrt{2013}+\sqrt{2011}}\)
\(=\frac{2}{\sqrt{2013}+\sqrt{2011}}\)
+ \(\sqrt{2012}-\sqrt{2010}=\frac{\left(\sqrt{2012}-\sqrt{2010}\right)\left(\sqrt{2012}+\sqrt{2010}\right)}{\sqrt{2012}+\sqrt{2010}}\)
\(=\frac{2}{\sqrt{2012}+\sqrt{2010}}\)
+ \(\sqrt{2013}+\sqrt{2011}>\sqrt{2012}+\sqrt{2010}\)
\(\Rightarrow\frac{2}{\sqrt{2013}+\sqrt{2011}}< \frac{2}{\sqrt{2012}+\sqrt{2010}}\)
\(\Rightarrow\sqrt{2013}-\sqrt{2011}< \sqrt{2012}-\sqrt{2010}\)
Ta có:\(\) \(\left(\sqrt{2012}-\sqrt{2011}\right)\left(\sqrt{2012}+\sqrt{2011}\right)=1\)
\(\left(\sqrt{2011}-\sqrt{2010}\right)\left(\sqrt{2011}+\sqrt{2010}\right)=1\)
Vì \(\left(\sqrt{2012}+\sqrt{2011}\right)>\left(\sqrt{2011}+\sqrt{2010}\right)\)
nên \(\left(\sqrt{2012}-\sqrt{2011}\right)< \left(\sqrt{2011}-\sqrt{2010}\right)\)
Vậy A<B.
Chưa tính nhưg nghĩ là
\(\sqrt{2012}-\sqrt{2011}\) > \(\sqrt{2011}-\sqrt{2010}\)