Cho S=1/2-1/3+1/4-1/5+1/6-1/7+...+1/48-1/49
Chứng minh 1/5<S<2/5
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![](https://rs.olm.vn/images/avt/0.png?1311)
5: \(=\dfrac{43}{8}-\dfrac{19}{10}=\dfrac{430}{80}-\dfrac{152}{80}=\dfrac{278}{80}=\dfrac{139}{40}\)
6: \(=\dfrac{-13}{4}-\dfrac{7}{3}=\dfrac{-39}{12}-\dfrac{28}{12}=\dfrac{-67}{12}\)
7: \(=\dfrac{-41}{8}+\dfrac{14}{4}=\dfrac{-41}{8}+\dfrac{28}{8}=\dfrac{-13}{8}\)
8: \(=\dfrac{-43}{7}+\dfrac{-43}{6}=\dfrac{-559}{42}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
5, \(5\dfrac{3}{8}-1\dfrac{9}{10}=\dfrac{139}{40}\)
6, \(\left(-3\dfrac{1}{4}\right)+\left(-2\dfrac{1}{3}\right)=-\dfrac{67}{12}\)
7, \(\left(-5\dfrac{1}{8}\right)+3\dfrac{2}{4}=-\dfrac{13}{8}\)
8, \(\left(-6\dfrac{1}{7}\right)+\left(-7\dfrac{1}{6}\right)=-\dfrac{559}{42}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
1) 1/3 x 1/2 x 3/7 = 1/6 x 3/7 = 1/14
2) 5/4 x 1/3 + 1/7 = 5/12 + 1/7 = 47/84
3) 8 x (8/9 - 2/3) = 8 x 2/9 = 16/9
4) 5/6 x 48/20 x 1/2 = 2 x 1/2 = 1
5) (2/5 + 3/4) x 8 = 23/20 x 8 = 46/5
6) 10 x (1/2 - 1/5) = 10 x 3/10 = 3
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
1) 1/3 x 1/2 x 3/7 = 3/42 = 1/14
2) 5/4 x 1/3 +1/7 = 5/12 + 1/7 = 35/84 + 12/84 = 47/84
3) 8 x ( 8/9 - 2/3 ) = 8 x 2/9 = 16/9
4) 5/6 x 48/20 x 1/2 = 240/240 = 1
5) ( 2/5 + 3/4 ) + 8 = 23/20 + 8 = 23//20 + 160/20 = 183/20
6) 10 x ( 1/2 - 1/5 ) = 10 x 3/10 = 10/1 x 3/10 = 30/10 = 3
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Tính các tổng sau:
1, S=1-2+3_4+..+25-26
S =-1+3-5+7-...-53+55 ( có 28 số hạng )
= (-1+3)+(-5+7)+...+(-53+55) ( có 28:2=14 nhóm )
= 2+2+...+2
= 2 . 14
= 28
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\dfrac{4}{7}-\dfrac{1}{3}< \dfrac{1}{2}-\dfrac{2}{9}\)
\(\dfrac{125}{100}-\dfrac{48}{64}< \dfrac{6}{7}-\dfrac{14}{28}\)
\(\dfrac{7}{5}-\dfrac{22}{25}=\dfrac{1}{5}+\dfrac{4}{10}\)\(\dfrac{45}{75}+\dfrac{22}{33}>\dfrac{4}{5}+\dfrac{4}{6}\)
\(S=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{48}-\frac{1}{49}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\left(\frac{1}{6}-\frac{1}{7}\right)+...+\left(\frac{1}{48}-\frac{1}{49}\right)\)
\(>\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}=\frac{13}{60}>\frac{12}{60}=\frac{1}{5}\)
\(S=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{48}-\frac{1}{49}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\left(-\frac{1}{7}+\frac{1}{8}\right)+...+\left(-\frac{1}{47}+\frac{1}{48}\right)-\frac{1}{49}\)
\(< \frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}=\frac{23}{60}< \frac{24}{60}=\frac{2}{5}\)