Ta có:

1/3 + 1/6 + 1/10 + ... + 1/x(x+1):2 = 2001/2003

=> 2/6 + 2/12 + 2/20 + ... + 2/x(x+1) = 2001/2003

=> 2 [1/6 + 1/12 + 1/20 + ... + 1/x(x+1)] = 2001/2003

=> 2 [1/2x3 + 1/3x4 + 1/4x5 + ... + 1/x+(x+1)] = 2001/2003

=> 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/x - 1/x+1= 2001/2003 : 2

=> 1/2 - 1/x+1 = 2001/4006

=> 1/x+1 = 1/2 - 2001/4006 = 1/2003

=> x+1 = 2003 = 2002 + 1 

=>x = 2002

\(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{4020}{2011}\)

\(\Rightarrow\dfrac{2}{6}+\dfrac{2}{12}+\dfrac{2}{20}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{4020}{2011}\)

\(\Rightarrow\dfrac{2}{2.3}+\dfrac{2}{3.4}+\dfrac{2}{4.5}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{4020}{2011}\)

\(\Rightarrow2\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{x\left(x+1\right)}\right)=\dfrac{4020}{2011}\)

\(\Rightarrow\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{4020}{2011}:2\)

\(\Rightarrow\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{2010}{2011}\)

\(\Rightarrow\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{2010}{2011}\)

\(\Rightarrow\dfrac{1}{x+1}=\dfrac{1}{2}-\dfrac{2010}{2011}\)

\(\Rightarrow\dfrac{1}{x+1}=-\dfrac{2009}{4022}\)

\(\Rightarrow4022=-2009\left(x+1\right)\)

\(\Rightarrow4022=-2009x-2009\)

\(\Rightarrow2009x=-2009-4022\)

\(\Rightarrow2009x=-6031\)

\(\Rightarrow x=-\dfrac{6031}{2009}\)

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{x\left(x+1\right)}=\frac{1999}{2001}\)

\(\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{1999}{2001}\)

\(2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{1999}{2001}\)

\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{1999}{2001}:2=\frac{1999}{4002}\)

\(\frac{1}{2}-\frac{1}{x+1}=\frac{1999}{4002}\)

\(\frac{1}{x+1}=\frac{1}{2}-\frac{1999}{2001}=\frac{1}{2001}\)

=> x + 1 = 2001

=> x = 2001 - 1

=> x = 2000

\(\Rightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+..+\frac{2}{x\left(x+1\right)}=\frac{1999}{2001}\)

\(2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+..+\frac{1}{x\left(x+1\right)}\right)=\frac{1999}{2001}\)

   \(\frac{1}{6}+\frac{1}{12}+..+\frac{1}{x\left(x+1\right)}=\frac{1999}{2001}:\frac{1}{2}\)

  \(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x\left(x+1\right)}=\frac{1999}{4002}\)

  \(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+..+\frac{1}{x}-\frac{1}{x+1}=\frac{1999}{4002}\)

\(\frac{1}{2}-\frac{1}{x+1}=\frac{1999}{4002}\)

      \(\frac{1}{x+1}=\frac{1}{2}-\frac{1999}{4002}\)

    \(\frac{1}{x+1}=\frac{1}{2001}\)

=> x + 1 = 2001

=> x =    2001 - 1

=> x = 2000 

(*) <=> 1\6 + 1\12 +.. + 1\x.(x+1) = 2009\(2011.2) 
ma 
1\2.3 =1\2-1\3 
1\3.4=1\3-1\4 
............... 
1\x(x+1)= 1\x-1\(x+1) 

cong tung ve ta dc 

Vt= 1\2- 1\(x+1) =2009\(2.2011) 

<=> 2011\(2.2011) -2009\(2.2011) =1\(x+1) 

<=> 1\2011 =1\(x+1) 

=> x=2010

1/3 + 1/6 + 1/10 + ... + 2/x(x+1) = 1999/2001

nhân 1/2 vào 2 vế ta được vế trái là :

\(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}=\frac{1}{2}.\frac{1999}{2001}\)

\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{1}{2}.\frac{1999}{2001}\)

\(\frac{1}{2}-\frac{1}{x+1}=\frac{1}{2}.\frac{1999}{2001}\)

\(\frac{x-1}{2.\left(x+1\right)}=\frac{1}{2}.\frac{1999}{2001}\)

\(\frac{x-1}{\left(x+1\right)}=\frac{1999}{2001}\)

suy ra : 2001x - 2001 = 1999x + 1999

2x = 1999 + 2001 = 4000

=> x = 2000