a) chứng minh\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}\)\(=\)\(\frac{1}{\sqrt{n}}\)\(-\)\(\frac{1}{\sqrt{n+1}}\)n\(\in\)zb) TínhA=...
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a) chứng minh
\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}\)\(=\)\(\frac{1}{\sqrt{n}}\)\(-\)\(\frac{1}{\sqrt{n+1}}\)n\(\in\)z
b) Tính
A= \(\sqrt{4+\sqrt{10+2\sqrt{5}}}\)\(+\)\(\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
B=\(\frac{1}{2\sqrt{1+1\sqrt{2}}}\)+\(\frac{1}{3\sqrt{2+2\sqrt{3}}}\)+\(\frac{1}{4\sqrt{3+3\sqrt{4}}}\)+...\(\frac{1}{100\sqrt{99+99\sqrt{100}}}\)
A.\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2n-n^2\left(n+1\right)}\) \(=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)\left(n+1-n\right)}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}\)
=\(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
b. ap dungtinh B =\(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}=1-\frac{1}{10}=\frac{9}{10}\)