\(\frac{\text{9 x999999999999999999x999999999999}}{\text{9 x999999999999999999x999999999999}}\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\)
= \(\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)
= \(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)
= \(\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{99.100}\right)=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)=\frac{1}{2}.\frac{4949}{9900}=\frac{4949}{19800}\left(\text{đpcm}\right)\)
\(VT=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\)
\(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)
\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)
\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{99.100}\right)=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{9900}\right)=\frac{1}{2}.\frac{4949}{9900}=\frac{4949}{19800}=VP\) (đpcm)
![](https://rs.olm.vn/images/avt/0.png?1311)
theo bài ra ta có
n = 8a +7=31b +28
=> (n-7)/8 = a
b= (n-28)/31
a - 4b = (-n +679)/248 = (-n +183)/248 + 2
vì a ,4b nguyên nên a-4b nguyên => (-n +183)/248 nguyên
=> -n + 183 = 248d => n = 183 - 248d (vì n >0 => d<=0 và d nguyên )
=> n = 183 - 248d (với d là số nguyên <=0)
vì n có 3 chữ số lớn nhất => n<=999 => d>= -3 => d = -3
=> n = 927
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\left(\frac{x-3\sqrt{x}}{x-9}-1\right)\left(\frac{9-x}{x+\sqrt{x}-6}+\frac{\sqrt{x}-3}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}+3}\right)\)
\(=\left(\frac{\sqrt{x}}{\sqrt{x}+3}-1\right)\left(\frac{9-x+x-9-x+4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}\right)\)
\(=\frac{-3}{\sqrt{x}+3}.\frac{2-\sqrt{x}}{\sqrt{x}+3}\)
\(=\frac{3\sqrt{x}-6}{\left(\sqrt{x}+3\right)^2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(=3\frac{2}{9}-2\frac{7}{9}=\frac{29}{9}-\frac{25}{9}=\frac{4}{9}\)
Chúc bạn học tốt!!!
\(3\frac{2}{9}\)\(-\left(1\frac{5}{9}+1\frac{2}{9}\right)\)
= \(\frac{29}{9}\)\(-\left(\frac{14}{9}+\frac{11}{9}\right)\)
= \(\frac{29}{9}\)\(-\frac{25}{9}\)
= \(\frac{4}{9}\)