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A=2001/2002+2002/2003
B=2001/2002+2003+2002/2002+2003
(tớ tách B ra đấy)
mà 2001//2002+2002/2003>2001/2002+2003+ 202/2002+2003
A>B
![](https://rs.olm.vn/images/avt/0.png?1311)
ta có \(\frac{2000+2002}{2001+2003}\)= \(\frac{2000}{2001+2003}\)+ \(\frac{2002}{2001+2003}\)=\(\frac{2000}{4004}\)+\(\frac{2002}{4004}\)
ta có \(\frac{2000}{2001}\)>\(\frac{2000}{4004}\) và \(\frac{2002}{2003}\)> \(\frac{2002}{4004}\)
nên \(\frac{2000}{2001}\)+\(\frac{2002}{2003}\)>\(\frac{2000}{4004}\)+\(\frac{2002}{4004}\)
vậy \(\frac{2000}{2001}\)+\(\frac{2002}{2003}\)>\(\frac{2000+2002}{2001+2003}\)
\(\frac{2000+2002}{2001+2003}=\frac{2000}{2001+2003}+\frac{2002}{2001+2003}< \frac{2000}{2001}+\frac{2002}{2003}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
2001/2002=1-1/2002
2002/2003=1-1/2003
vi 1/2003<1/2002 nen 2001/2002<2002/2003
Ta có: 2003 x 2001 < 2002 x 2002
=> \(\frac{2001}{2002}\)<\(\frac{2002}{2003}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
ta thấy:
\(B< 1\Rightarrow B< \frac{10^{2002}+1+9}{10^{2003}+1+9}=\frac{10^{2002}+10}{10^{2003}+10}=\frac{10\left(10^{2001}+1\right)}{10\left(10^{2002}+1\right)}=\frac{10^{2001}+1}{10^{2002}+1}=A\)
=>B<A
vậy.......
Ta có:
\(A=\frac{10^{2001}+1}{10^{2002}+1}\Rightarrow10A=\frac{10\left(10^{2001}+1\right)}{10^{2002}+1}=\frac{10^{2002}+10}{10^{2002}+1}=\frac{10^{2002}+1+9}{10^{2002}+1}=1+\frac{9}{10^{2002}+1}\)
\(B=\frac{10^{2002}+1}{10^{2003}+1}\Rightarrow10B=\frac{10\left(10^{2002}+1\right)}{10^{2003}+1}=\frac{10^{2003}+10}{10^{2003}+1}=\frac{10^{2003}+1+9}{10^{2003}+1}=1+\frac{9}{10^{2003}+1}\)
Vì \(\frac{9}{10^{2002}+1}>\frac{9}{2^{2003}+1}\Rightarrow1+\frac{9}{10^{2002}+1}>1+\frac{9}{2^{2003}+1}\Rightarrow10A>10B\Rightarrow A>B\)
Vậy A > B
![](https://rs.olm.vn/images/avt/0.png?1311)
=1+1/2001+1+1/2002+1+1/2003+...+1+1/2008=8+1/2001+1/2002+1/2003+...+1/2008>8
\(\frac{2002}{2001}+\frac{2003}{2002}+\frac{2004}{2003}+\frac{2005}{2004}+\frac{2006}{2005}+\frac{2007}{2006}+\frac{2008}{2007}+\frac{2009}{2008}>8\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\dfrac{2001+2002}{2002+2003}< \dfrac{2001}{2002}+\dfrac{2002}{2003}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\frac{10^{2001}+1}{10^{2002}+1}=\frac{\left(10^{2001}+1\right)\left(10^{2003}+1\right)}{\left(10^{2002}+1\right)\left(10^{2003}+1\right)}=\frac{10^{4004}+10^{2001}+10^{2003}+1}{\left(10^{2002}+1\right)\left(10^{2003}+1\right)}\)
\(B=\frac{10^{2002}+1}{10^{2003}+1}=\frac{\left(10^{2002}+1\right)\left(10^{2002}+1\right)}{\left(10^{2003}+1\right)\left(10^{2002}+1\right)}=\frac{10^{4004}+2.10^{2002}+1}{\left(10^{2003}+1\right)\left(10^{2002}+1\right)}\)
Vì 102001 + 102003 < 2.102002 nên A < B
![](https://rs.olm.vn/images/avt/0.png?1311)
minh lam duoc roi . cach viet phan so ban bam vao o mau vang o cuoi trang .cu di con chuot xuong cuoi trang thi thay 1 o vang , vao xem huong dan la biet ngay ma.
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Cách 1 :
Ta có :
2001/2002 = 4008003/4010006 ( quy đồng mẫu số )
2002/2003=4008004/4010006 ( quy đồng mẫu số )
Vì phân số 4008004/4010006 > 4008003/4010006 nên phân số 2001/2002 < 2002/2003
Cách 2 :
2001/2002 = 4006002/4008004 ( quy đồng tử số )
2002/2003 = 4006002/4008003 ( quy đồng tử số )
Vì 4006002/4008004 < 4006002/4008003 nên phân số 2001/2002 < 2002/2003
Cách 3 :
2001/2002 = 1 - 1/2002
2002/2003 = 1 - 1/2003
Vì 1/2002 > 1/2003 nên 2001/2002 < 2002/2003