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

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

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

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

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

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

=> 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

1/2x[1/3+1/6+1/10+....2/x.(x+1)]=1999/2001x1/2

1/2x3+1/3x4+....+1/x(x+1)=1999/4002

1/2-1/3+1/3-1/4+....+1/x-1/x+1=1999/4002

1/2-1/x+1=1999/4002

1/x+1=1/2-1999/4002

1/x+1=1/2001

=>x+1=2001

x=2001-1

x=2000

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{1999}{2001}\)

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

\(\Rightarrow2\left(\frac{1}{6}+\frac{1}{12}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{1999}{2001}\)

\(\Rightarrow2\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{1999}{2001}\)

\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{1999}{2001}\)

\(\Rightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{1999}{2001}\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{1999}{2001}:2\)

\(\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{1999}{2001}:2=\frac{1}{2001}\Rightarrow x+1=2001\Rightarrow x=2000\)

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