\(\sqrt{\dfrac{1}{3}+\dfrac{1}{2\sqrt{3}}}+\sqrt{\dfrac{1}{3}-\dfrac{1}{2\sqrt{3}}}\)
\(=\sqrt{\dfrac{2+\sqrt{3}}{6}}+\sqrt{\dfrac{2-\sqrt{3}}{6}}\)
\(=\sqrt{\dfrac{4+2\sqrt{3}}{12}}+\sqrt{\dfrac{4-2\sqrt{3}}{12}}\)
\(=\sqrt{\dfrac{\left(\sqrt{3}+1\right)^2}{12}}+\sqrt{\dfrac{\left(\sqrt{3}-1\right)^2}{12}}\)
\(=\dfrac{\sqrt{3}+1}{2\sqrt{3}}+\dfrac{\sqrt{3}-1}{2\sqrt{3}}\)
\(=1\)
1) Chứng minh rằng: \(1+\dfrac{1}{2\sqrt{2}}+\dfrac{1}{3\sqrt{3}}+...+\dfrac{1}{n\sqrt{n}}< 2\sqrt{2}\left(n\in N\right)\)
2) Chứng minh rằng: \(\dfrac{2}{3}+\sqrt{n+1}< 1+\sqrt{2}+\sqrt{3}+...+\sqrt{n}< \dfrac{2}{3}\left(n+1\right)\sqrt{n}\)
3) \(2\sqrt{n}-3< \dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+...+\dfrac{1}{\sqrt{n}}< 2\sqrt{n}-2\)
4) \(\dfrac{\sqrt{2}-\sqrt{1}}{2+1}+\dfrac{\sqrt{3}-\sqrt{2}}{3+2}+...+\dfrac{\sqrt{n+1}-\sqrt{n}}{n+1+n}< \dfrac{1}{2}\left(1-\dfrac{1}{\sqrt{n+1}}\right)\)
tính:
\(A=\dfrac{1+\dfrac{\sqrt{3}}{2}}{1+\sqrt{1+\dfrac{\sqrt{3}}{2}}}+\dfrac{1-\dfrac{\sqrt{3}}{2}}{1-\sqrt{1-\dfrac{\sqrt{3}}{2}}}\)
thực hiện phép tính sau: \(\dfrac{\sqrt{3}+\sqrt{2}-1}{2+\sqrt{6}}+\dfrac{\sqrt{2}-\sqrt{3}}{\sqrt{2}+1}\left(\dfrac{\sqrt{3}}{2-\sqrt{6}}+\dfrac{\sqrt{3}}{2+\sqrt{6}}\right)-\dfrac{1}{\sqrt{2}}\)
CMR:
\(\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{3}}+....+\dfrac{1}{\left(n+1\right)\left(\sqrt{n}+n\sqrt{n+1}\right)}< 1\)
\(\sqrt{\dfrac{1}{3}+\dfrac{1}{2\sqrt{3}}}- \sqrt{\dfrac{1}{3}\dfrac{1}{2\sqrt{3}}}\)
1)tính
a)\(\left(\dfrac{1}{5}\sqrt{500}-3\sqrt{45}+5\sqrt{20}\right):\sqrt{5}\)
b)\(\left(\dfrac{\sqrt{3}+1}{\sqrt{3}-1}-\dfrac{\sqrt{3}-1}{\sqrt{3}+1}\right).\sqrt{\dfrac{1}{48}}\)
c)\(\left(\dfrac{2\sqrt{3}+3}{\sqrt{3}+2}+\dfrac{2\sqrt{2}}{\sqrt{2}+1}\right):\left(\sqrt{12}+\sqrt{18}\right)\)
So sánh A với 1
\(A=\dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+\dfrac{1}{4\sqrt{3}+3\sqrt{4}}+...+\dfrac{1}{100\sqrt{99}+99\sqrt{100}}\)
CM: \(\left(\dfrac{2}{\sqrt{6}-1}+\dfrac{3}{\sqrt{6}-2}+\dfrac{3}{\sqrt{6}-3}\right).\dfrac{5}{9\sqrt{6}+4}=\dfrac{1}{2}\)
\(\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\dfrac{1}{\sqrt{3}+\sqrt{4}}+\dfrac{1}{\sqrt{4}+\sqrt{5}}+...+\dfrac{1}{\sqrt{99}+\sqrt{100}}\)
Đề là rút gọn
