\(A=\frac{1}{2003.2002}-\frac{1}{2002.2001}-\frac{1}{2001.2000}-....-\frac{1}{3.2}-\frac{1}{2.1}\)

    \(=-\left(\frac{1}{2003.2002}+\frac{1}{2002.2001}+\frac{1}{2001.2000}+...+\frac{1}{3.2}+\frac{1}{2.1}\right)\)

     \(=-\left(\frac{1}{2003}-\frac{1}{2002}+\frac{1}{2002}-\frac{1}{2001}+...+\frac{1}{3}-\frac{1}{2}+\frac{1}{2}-1\right)\)

      \(=-\left(\frac{1}{2003}-1\right)=-\left(-\frac{2002}{2003}\right)=\frac{2002}{2003}\)

Vậy ....

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\(\frac{1}{2003.2002}-\frac{1}{2002.2001}-...-\frac{1}{2.1}\)

\(=\frac{1}{2003.2002}-\left(\frac{1}{1.2}+\frac{1}{3.2}+...+\frac{1}{2001.2002}\right)\)

\(=\frac{1}{2003.2002}-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...+\frac{1}{2000}-\frac{1}{2001}+\frac{1}{2001}-\frac{1}{2002}\right)\)

\(=\frac{1}{2003.2002}-\left(1-\frac{1}{2002}\right)\)

\(=\frac{1}{2003.2002}-\frac{2001}{2002}\)

\(\frac{1}{2003.2002}-\frac{1}{2002.2001}-...-\frac{1}{2.1}\)

\(=\frac{1}{2003.2002}-\left(\frac{1}{2002.2001}+\frac{1}{2001.2000}+...+\frac{1}{2.1}\right)\)

\(=\frac{1}{2003.2002}-\left(\frac{1}{2002}-\frac{1}{2001}+\frac{1}{2001}-\frac{1}{2000}+...+\frac{1}{2}-1\right)\)

\(=\frac{1}{2003.2002}-\left(\frac{1}{2002}-1\right)\)

\(=\frac{1}{2003.2002}-\left(\frac{1}{2002}-\frac{2002}{2002}\right)\)

\(=\frac{1}{2003.2002}-\frac{-2001}{2002}\)

\(=\frac{1}{2003}-\frac{1}{2002}+\frac{2001}{2002}\)

\(=\frac{1}{2003}+\frac{2000}{2002}\)

\(=\frac{1}{2003}+\frac{1000}{1001}\)

Ko chắc ạ! Bạn nào rãnh thì check hoặc nhận xét hộ!

\(\dfrac{1}{2003.2002}-\dfrac{1}{2002.2001}-...-\dfrac{1}{3.2}-\dfrac{1}{2.1}\)

= \(\dfrac{1}{2003.2002}-\left(\dfrac{1}{2002.2001}+...+\dfrac{1}{3.2}+\dfrac{1}{2.1}\right)\)

= \(\dfrac{1}{2003.2002}-\left(\dfrac{1}{2002}-\dfrac{1}{2001}+...+\dfrac{1}{3}-\dfrac{1}{2}+\dfrac{1}{2}-1\right)\)

= \(\dfrac{1}{2003.2002}-\dfrac{1}{2002}+1\)

= \(\dfrac{1-2003+2003.2002}{2003.2002}\)

= \(1-\dfrac{2002}{2003.2002}=1-\dfrac{1}{2003}\) = \(\dfrac{2002}{2003}\)

\(-\frac{1}{2003\cdot2002}-\frac{1}{2002\cdot2001}-\frac{1}{2001\cdot2000}-...-\frac{1}{2\cdot1}\) 

\(=-1\left(\frac{1}{1\cdot2}+...+\frac{1}{2000\cdot2001}+\frac{1}{2001\cdot2002}+\frac{1}{2002\cdot2003}\right)\) 

\(=-1\left(\frac{1}{1}-\frac{1}{2}+...+\frac{1}{2000}-\frac{1}{2001}+\frac{1}{2001}-\frac{1}{2002}+\frac{1}{2002}-\frac{1}{2003}\right)\) 

\(=-1\left(1-\frac{1}{2003}\right)\) 

\(=-1\left(\frac{2003}{2003}-\frac{1}{2003}\right)\)              

\(=-1\cdot\frac{2002}{2003}\) 

\(=-\frac{2002}{2003}\)

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