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Ta có : \(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{1599}{1600}\)
\(=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{6}\right)...\left(1-\frac{1}{1600}\right)\)
Đặt \(B=\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{1600}{1601}\)
\(=\left(1-\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{7}\right)...\left(1-\frac{1}{1601}\right)\)
Vì \(\frac{1}{2}>\frac{1}{3};\frac{1}{4}>\frac{1}{5};\frac{1}{6}>\frac{1}{7};...;\frac{1}{1600}>\frac{1}{1601}\)
\(\Rightarrow1-\frac{1}{2}< 1-\frac{1}{3};1-\frac{1}{4}< 1-\frac{1}{5};1-\frac{1}{6}< 1-\frac{1}{7};...;1-\frac{1}{1600}< 1-\frac{1}{1601}\)
\(\Rightarrow A< B\)
hay A<\(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{1600}{1601}\)
Vậy A<\(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{1600}{1601}\).
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta luôn có:
\(\frac{1}{2}< \frac{2}{3}\)
\(\frac{3}{4}< \frac{4}{5}\)
\(\frac{5}{7}< \frac{6}{7}\)
\(........\)
\(\frac{1599}{1600}< \frac{1600}{1601}\)
Từ trên: \(\Rightarrow A=\frac{1}{2}.\frac{3}{4}....\frac{1599}{1600}\left(1\right)\)
\(\Rightarrow\frac{1}{2}.\frac{3}{4}...\frac{1599}{1600}< \frac{2}{3}.\frac{4}{5}....\frac{1600}{1601}\left(2\right)\)
Từ: \(\left(1\right)\left(2\right)\Rightarrow A< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{1600}{1601}\left(đpcm\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(=\frac{2.4.3.5.4.6.5.7....39.41}{3.4.5....40.3.4.5....40}=\frac{\left(2.3.4.....39\right).\left(4.5.6.....41\right)}{\left(3.4.5......40\right).\left(3.4.5......40\right)}=\frac{2.3.4.....39}{3.4.5....40}.\frac{4.5.6.....41}{3.4.5.....40}=\frac{2}{40}.\frac{41}{3}=\frac{41}{60}\)
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2) \(\left(\dfrac{1}{6}+2\dfrac{1}{2}-10,75\right)\cdot x-7=\left(\dfrac{2}{5}+\dfrac{3}{8}+0,025\right):0,1\)
\(\left(\dfrac{1}{6}+\dfrac{5}{2}-\dfrac{43}{4}\right)x-7=\left(\dfrac{2}{5}+\dfrac{3}{8}+\dfrac{1}{40}\right):\dfrac{1}{10}\)
\(-\dfrac{97}{12}x-7=\dfrac{4}{5}\cdot10\)
\(-\dfrac{97}{12}x=8+7\\ -\dfrac{97}{12}x=15\\ x=15:\left(-\dfrac{97}{12}\right)\\ x=-\dfrac{180}{97}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có
\(\frac{1}{2}< \frac{2}{3},\frac{3}{4}< \frac{4}{5},...,\frac{1599}{1600}< \frac{1600}{1601}\)
Do đó ta có
A=\(\frac{1}{2}\times\frac{3}{4}\times...\times\frac{1599}{1600}< \frac{2}{3}\times\frac{4}{5}\times...\times\frac{1600}{1601}\)
#Châu's ngốc