Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Tính
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{899}+\sqrt{900}}\)
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+.....+\frac{1}{\sqrt{899}+\sqrt{900}}\)
=\(\frac{\sqrt{1}-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+....+\frac{\sqrt{899}-\sqrt{900}}{899-900}\)
\(=\frac{\sqrt{1}-\sqrt{2}}{-1}+\frac{\sqrt{2}-\sqrt{3}}{-1}+....+\frac{\sqrt{899}-\sqrt{900}}{-1}\)
\(=\frac{\sqrt{1}-\sqrt{2}+\sqrt{2}-\sqrt{3}+....+\sqrt{899}-\sqrt{900}}{-1}\)
\(=\frac{\sqrt{1}-\sqrt{900}}{-1}\)
\(=\frac{1-30}{-1}=\frac{-29}{-1}=29\)
=
\(\sqrt{14+\sqrt{16900}}-\sqrt{19+\sqrt{900}}+\sqrt{45+\sqrt{3025}}\)
\(=\sqrt{14+\sqrt{130^2}}-\sqrt{19+\sqrt{30^2}}+\sqrt{45+\sqrt{55^2}}\)
\(=\sqrt{14+130}-\sqrt{19+30}+\sqrt{45+55}\)
\(=\sqrt{144}-\sqrt{49}+\sqrt{100}\)
\(=\sqrt{12^2}-\sqrt{7^2}+\sqrt{10^2}\)
\(=12-7+10\)
\(=5+10\)
\(=15\)
Xét \(a_n=\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2n-n^2\left(n+1\right)}\)
\(=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}=\frac{\sqrt{n}}{n}-\frac{\sqrt{n+1}}{n+1}\)
\(\Rightarrow S=\frac{\sqrt{1}}{1}-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{3}+...+\frac{\sqrt{899}}{899}-\frac{\sqrt{900}}{900}\)
\(S=1-\frac{\sqrt{900}}{900}=1-\frac{1}{30}=\frac{29}{30}\)
\(a,\sqrt{81}=9\)
\(b.\sqrt{8100}=90\)
\(c,\sqrt{64}=8\)
\(d,\sqrt{25}=5\)
\(e,\sqrt{0,64}=0,8\)
\(f,\sqrt{10000}=100\)
\(g,\sqrt{0,01}=0,1\)
\(h,\sqrt{\frac{49}{100}}=\frac{7}{10}\)
\(i,\sqrt{\frac{0,09}{121}}=\frac{0,3}{11}\)
\(j,\sqrt{\frac{4}{25}}=\frac{2}{5}\)
~Study well~
#JDW
a) 9
b) 90
c) 8
d) 5
e) 0,8
f) 100
g) 0,1
h) \(\frac{7}{10}\)
i) \(\frac{0,3}{11}\)
j) 0,4.
\(\sqrt{81}=9\)
\(\sqrt{0,64}=0,8\)
\(\sqrt{\frac{49}{100}}=\frac{7}{10}\)
\(\sqrt{8100}=90\)
\(\sqrt{100=}10\)
\(\sqrt{0,01}=0,1\)
\(\sqrt{\frac{4}{25}}=\frac{2}{5}\)
\(\sqrt{\frac{0,09}{121}}=\frac{0,3}{11}\)
\(\sqrt{81}=9\);\(\sqrt{0,64}=0,8\);\(\sqrt{\frac{49}{100}}=\frac{7}{10}\);\(\sqrt{8100}=90\); \(\sqrt{100}=10\); \(\sqrt{0,01}=0,1\); \(\sqrt{\frac{4}{25}}=\frac{2}{5}\); \(\sqrt{\frac{0,09}{121}}=\frac{0,3}{11}=\frac{3}{110}\)
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{899}+\sqrt{900}}\)
\(=\frac{\sqrt{2}-\sqrt{1}}{\left(\sqrt{2}-\sqrt{1}\right)\left(\sqrt{2}+\sqrt{1}\right)}+\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}+...+\frac{\sqrt{900}-\sqrt{899}}{\left(\sqrt{900}-\sqrt{899}\right)\left(\sqrt{900}+\sqrt{899}\right)}\)
\(=\frac{\sqrt{2}-1}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+...+\frac{\sqrt{900}-\sqrt{899}}{900-899}\)
\(=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{3}+...+\sqrt{900}-\sqrt{899}\)
\(=\sqrt{900}-\sqrt{1}\)
\(=30-1\)
\(=29\)
\(\sqrt{8100=90}\) \(\sqrt{3136=56}\)
\(\sqrt{3364=58}\) \(\sqrt{722500=850}\)
\(\sqrt{900=30}\)
\(\sqrt{3969=63}\)