rút gọn:\(\frac{1}{\sqrt{1}-\sqrt{2}}+\frac{1}{\sqrt{2}-\sqrt{3}}+...+\frac{1}{\sqrt{24}-\sqrt{25}}\)
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Rút gọn : \(A=\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}-...-\frac{1}{\sqrt{24}-\sqrt{25}}\)
Rút gọn:
\(\frac{1}{\sqrt{1}\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{24}\sqrt{25}}\)
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{24}+\sqrt{25}}\)
\(=\frac{\left(\sqrt{2}-\sqrt{1}\right)\left(\sqrt{2}+\sqrt{1}\right)}{\sqrt{1}+\sqrt{2}}+\frac{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}{\sqrt{2}+\sqrt{3}}\)
\(+...+\frac{\left(\sqrt{25}-\sqrt{24}\right)\left(\sqrt{25}+\sqrt{24}\right)}{\sqrt{24}+\sqrt{25}}\)
\(=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{25}-\sqrt{24}\)
\(=\sqrt{25}-1=5-1=4\)
\(\frac{1}{\sqrt{1}\sqrt{2}}+\frac{1}{\sqrt{2}\sqrt{3}}+...+\frac{1}{\sqrt{24}\sqrt{25}}\)
\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{24}}-\frac{1}{\sqrt{25}}\)
\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{25}}\)
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Rút gọn:
A = \(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{n-1}+\sqrt{n}}\)
B = \(\frac{1}{\sqrt{1}-\sqrt{2}}+\frac{1}{\sqrt{2}-\sqrt{3}}+...+\frac{1}{\sqrt{24}-\sqrt{25}}\)
\(A=\frac{\sqrt{2}-\sqrt{1}}{\left(\sqrt{2}-\sqrt{1}\right)\left(\sqrt{2}+\sqrt{1}\right)}+\frac{\sqrt{3}-\sqrt{2}}{\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{3}+\sqrt{2}\right)}+...+\frac{\sqrt{n}-\sqrt{n-1}}{\left(\sqrt{n}-\sqrt{n-1}\right)\left(\sqrt{n}+\sqrt{n-1}\right)}\)\(A=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+...+\sqrt{n}-\sqrt{n-1}\)
\(A=\sqrt{n}-\sqrt{1}\)
\(B=\frac{\sqrt{1}+\sqrt{2}}{\left(\sqrt{1}-\sqrt{2}\right)\left(\sqrt{1}+\sqrt{2}\right)}+\frac{\sqrt{2}+\sqrt{3}}{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}+...+\frac{\sqrt{24}+\sqrt{25}}{\left(\sqrt{24}-\sqrt{25}\right)\left(\sqrt{24}+\sqrt{25}\right)}\)
\(B=-\left(\sqrt{1}+\sqrt{2}\right)-\left(\sqrt{2}+\sqrt{3}\right)-...-\sqrt{24}+\sqrt{25}\)
\(B=-1-2\sqrt{2}-2\sqrt{3}-...-\sqrt{24}-5\)
\(B=-1-2\sqrt{2}-2\sqrt{3}-...-\sqrt{24}-5\)
\(B=-6-2\sqrt{2}-2\sqrt{3}-...-2\sqrt{24}\)
ta có \(\frac{1}{\sqrt{1}+\sqrt{2}}=\frac{\sqrt{1}-\sqrt{2}}{\left(\sqrt{1}+\sqrt{2}\right)\left(\sqrt{1}-\sqrt{2}\right)}=\frac{\sqrt{1}-\sqrt{2}}{1-2}=\sqrt{1}-\sqrt{2}\)
mấy cái kia cũng thế a
\(=>A=\left(\sqrt{2}-1\right)+\left(\sqrt{3}-2\right)+...+\left(\sqrt{n}-\sqrt{n-1}\right)\)=>A= căn n -1
Rút gọn:
\(\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{24}+\sqrt{25}}\)
chia ra là có quy luật
rồi rút gọn thành -1+5=4
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Rút gọn biểu thức sau:
\(A=\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-...-\frac{1}{\sqrt{24}-\sqrt{25}}\)
\(A=\frac{\sqrt{1}+\sqrt{2}}{1-2}-\frac{\sqrt{2}+\sqrt{3}}{2-3}+\frac{\sqrt{3}+\sqrt{4}}{3-4}-...-\frac{\sqrt{24}+\sqrt{25}}{24-25}\)
\(=-\sqrt{1}-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{3}-\sqrt{4}+...+\sqrt{24}+\sqrt{25}\)
\(=-\sqrt{1}+\sqrt{25}\)
\(=-1+5\)
\(=4.\)
Rút gọn:
\(\frac{1}{\sqrt{1}+\sqrt{2}}\)+\(\frac{1}{\sqrt{2}+\sqrt{3}}\)+...+\(\frac{1}{\sqrt{24}+\sqrt{25}}\)
Ta có:
\(A=\frac{1}{\sqrt{2}+\sqrt{1}}+\frac{1}{\sqrt{3}+\sqrt{2}}+...+\frac{1}{\sqrt{25}+\sqrt{24}}\)
\(=\frac{\sqrt{2}-\sqrt{1}}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+...+\frac{\sqrt{25}-\sqrt{24}}{25-24}\)
\(=\frac{\sqrt{2}-\sqrt{1}}{1}+\frac{\sqrt{3}-\sqrt{2}}{1}+...+\frac{\sqrt{25}-\sqrt{24}}{1}\)
\(=5-1=4\)
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rút gọn:
\(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+....+\frac{1}{25\sqrt{24}+24\sqrt{25}}\)
\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2n-n^2\left(n+1\right)}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
\(\Rightarrow\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{25\sqrt{24}+25\sqrt{24}}\)
\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{24}}-\frac{1}{\sqrt{25}}\)
\(=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{25}}=1-\frac{1}{5}=\frac{4}{5}\)
Rút gọn các biểu thức,a A frac{1}{sqrt{1}+sqrt{2}} + frac{1}{sqrt{2}+sqrt{3}}+ frac{1}{sqrt{3}+sqrt{4}}+ ......... + frac{1}{sqrt{n-1}+sqrt{n}}b B frac{1}{sqrt{1}-sqrt{2}}- frac{1}{sqrt{2}-sqrt{3}}- frac{1}{sqrt{3}-sqrt{4}}- .......... - frac{1}{sqrt{24}-sqrt{25}}
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Rút gọn các biểu thức,
a> A= \(\frac{1}{\sqrt{1}+\sqrt{2}}\) + \(\frac{1}{\sqrt{2}+\sqrt{3}}\)+ \(\frac{1}{\sqrt{3}+\sqrt{4}}\)+ ......... + \(\frac{1}{\sqrt{n-1}+\sqrt{n}}\)
b> B= \(\frac{1}{\sqrt{1}-\sqrt{2}}\)- \(\frac{1}{\sqrt{2}-\sqrt{3}}\)- \(\frac{1}{\sqrt{3}-\sqrt{4}}\)- .......... - \(\frac{1}{\sqrt{24}-\sqrt{25}}\)
\(A=\frac{\sqrt{2}-\sqrt{1}}{\left(\sqrt{2}-\sqrt{1}\right)\left(\sqrt{2}+\sqrt{1}\right)}+.......+\frac{\sqrt{n}-\sqrt{n-1}}{\left(\sqrt{n}-\sqrt{n-1}\right)\left(\sqrt{n}+\sqrt{n}-1\right)}\)
\(=\frac{\sqrt{2}-\sqrt{1}}{2-1}+........+\frac{\sqrt{n}-\sqrt{n-1}}{n-\left(n-1\right)}\)
\(=\sqrt{2}-\sqrt{1}+...........+\sqrt{n}-\sqrt{n-1}\)
\(=\sqrt{n}-\sqrt{1}=\sqrt{n}-1\)
bài B tương tự
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Rút gọn :
M = \(\frac{1}{3.\left(\sqrt{1}+\sqrt{2}\right)}+\frac{1}{5.\left(\sqrt{2}+\sqrt{3}\right)}+\frac{1}{7.\left(\sqrt{3}+\sqrt{4}\right)}+....+\frac{1}{49.\left(\sqrt{24}+\sqrt{25}\right)}\)