Question

So Sánh : \(\sqrt{2017^2-1}-\sqrt{2016^2-1}\) và \(\dfrac{2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)

Answer 1

\(\sqrt{2017^2-1}-\sqrt{2016^2-1}=\dfrac{2017^2-1-2016^2+1}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}=\dfrac{\left(2017-2016\right)\left(2017+2016\right)}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}=\dfrac{1+2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}>\dfrac{2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)

Answer 2

Ta có \(\left(\sqrt{2017^2-1}-\sqrt{2016^2-1}\right)\left(\sqrt{2017^2-1}+\sqrt{2016^2-1}\right)=2017^2-1+2016^2-1=2017^2-2016^2=\left(2017-2016\right)\left(2017+2016\right)=2017+2016< 2016+2016=2.2016\Rightarrow\left(\sqrt{2017^2-1}-\sqrt{2016^2-1}\right)\left(\sqrt{2017^2-1}+\sqrt{2016^2-1}\right)< 2.2016\Rightarrow\sqrt{2017^2-1}-\sqrt{2016^2-1}< \dfrac{2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)

Answer 3

mình bị lộn 2017+2016>2.2016\(\Rightarrow\sqrt{2017^2-1}-\sqrt{2016^2-1}< \dfrac{2.2016}{\sqrt{2017^2-1}+\sqrt{2016^2-1}}\)