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

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

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

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

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

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

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

000000000000000000000000000

bang 2000

hihi...sai do

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

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

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

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

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

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

=> 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 

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

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

\(\Rightarrow\frac{1}{6}+\frac{1}{12}+...+\frac{1}{x.\left(x+1\right)}=\frac{1999}{4002}\)

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

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

\(\Rightarrow\frac{1}{x+1}=\frac{1}{2001}\)

=> x=2000

Tìm stn biết: 1/3 + 1/6 + 1/10 + ...+2/x(x+1)=1999/2001

Bài giải: Gọi x là số tự nhiên cần tìm

Cho S= 1/3 + 1/6 +1/10 +...+ 1/x(x+1)

\(\Rightarrow\)S= 2/6 + 2/12+ 2/20 +...+ 2/2[x(x+1)]

\(\Rightarrow\)1/2S= 1/2.3 + 1/3.4 + 1/ 4.5 +...+1/2[x(x+1)]

\(\Rightarrow\)1/2S=1/2-1/3+1/3-1/4+...+1/(x-1) .(x+1)

\(\Leftrightarrow\)1/2S=1/2-1/x+1

Vì S = 1999 / 2001\(\Rightarrow\)1/2S=1/2-1 . (x+1)=1999/2001-1998-2001=1/2001

\(\Rightarrow\)1/x+1=1/2001

\(\Leftrightarrow\)x+1=2001

         x =2001-1 =2000

Vậy số tự nhiên đó là: 2000

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

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

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

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

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

\(\frac{x+1-2}{2\left(x+1\right)}=\frac{1999}{4002}\Rightarrow\frac{x-1}{2\left(x+1\right)}=\frac{1999}{4002}\Leftrightarrow4002\left(x-1\right)=1999.2\left(x+1\right)\)

=> 4002x - 4002 = 3998x + 3998

=> 4002x - 3998x = 3998 + 4002

=> 4x               = 8000

=> x                  = 2000

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

1/6+1/12+1/20+...+2/x(x+1)=1999/2001

2(1/6+1/12+1/20+...+1/x(x+1)=1999/2001

1/6+1/12+1/20+1/x(x+1)=1999/2001:2

1/6+1/12+1/20+...+1/x(x+1)=1999/4002

1/2x3+1/3x4+1/4x5+...+1/x(x+1)=1999/4002

1/2-1/3+1/3-1/4+1/4-1/5+...+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

Vậy x=2000.

làm thế nào để ra 2/6 và 2/12 vậy

Chưa chắc là đề sai!!!!!!!!!!!!!!

\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2001}{2003}\)

\(\Leftrightarrow\frac{1}{6}+\frac{1}{12}+...+\frac{1}{x\left(x+1\right)}=\frac{2001}{2003}.\frac{1}{2}=\frac{2001}{4006}\)

\(\Leftrightarrow\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...\frac{1}{x}-\frac{1}{x+1}=\frac{2001}{4006}\)

\(\Leftrightarrow\)\(\frac{1}{2}-\frac{1}{x+1}=\frac{2001}{4006}\)

\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2001}{4006}\)

\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2003}\)

\(x+1=2003\)

\(x=2002\)

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