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XO
5 tháng 1 2020
Ta có : \(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)
=> \(2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2011}{2013}\)
=> \(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}=\frac{2011}{4026}\)
=> \(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{2011}{4026}\)
=> \(\frac{1}{2}-\frac{1}{x+1}=\frac{2011}{4016}\Rightarrow\frac{1}{x+1}=\frac{1}{2013}\Rightarrow x+1=2013\Rightarrow x=2012\)
NT
2
17 tháng 9 2016
1/2x3/2+1/3x4/2+1/4x5/2+1/5x6/2+.......+2/Xx(X+1)=2011/2013
2/2x3+2/3x4+2/4x5+2/5x6+.....+2/Xx(X+1)=2011/2013
2x(1/2x3+1/3x4+1/4x5+1/5x6+....+1/Xx(x+1)=2011/2013
1/2x3+1/3x4+1/4x5+1/5x6+....+1/Xx(X+1)=2011/4026
1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+.....+ 1/x-1/x+1=2011/4026
1/2-1/x+1=2011/4026
1/x+1=1/2-2011/4026
1/x+1=1/2013
Suy ra x=2012
ES
1
DD
5 tháng 8 2016
\(=\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+........+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)
\(\Rightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+........+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)
\(=2.\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+........+\frac{1}{x.\left(x+1\right)}\right)=\frac{2011}{2013}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.........+\frac{1}{x}-\frac{1}{x+1}=\frac{2011}{2013}:2\)
\(=\frac{1}{2}-\frac{1}{x+1}=\frac{2011}{4026}\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2011}{4026}=\frac{1}{2013}\)
\(\Rightarrow x+1=2013\Rightarrow x=2012\)
DH
0
PY
1
23 tháng 8 2018
\(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{2011}{2013}\)
\(\Rightarrow2.\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{x\left(x+1\right)}\right)=\dfrac{2011}{2013}\)
\(\Rightarrow\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+1}=\dfrac{2011}{4026}\)
\(\Rightarrow\dfrac{1}{2}-\dfrac{1}{x+1}=\dfrac{2011}{4026}\)
\(\Rightarrow\dfrac{1}{x+1}=\dfrac{1}{2013}\)
\(\Rightarrow x+1=2013\)
\(\Rightarrow x=2012\)
Chúc hok dốt!
23 tháng 8 2018
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)
\(\Leftrightarrow\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)
\(\Rightarrow\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)
\(\Rightarrow2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
\(\Rightarrow2.\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{2011}{2013}:2\)
\(\Rightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{2011}{4026}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2}-\frac{2011}{4026}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{2013}\)
\(\Rightarrow x+1=2013\)
\(\Rightarrow x=2013-1\)
\(\Rightarrow x=2012\)
23 tháng 8 2018
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)
\(\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\) (1/3=2/6;...)
\(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)
\(2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
\(2.\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
\(1-\frac{2}{x+1}=\frac{2011}{2013}\)
\(\frac{2}{x+1}=\frac{2}{2013}\)
=> x + 1 = 2013
x = 2012
DS
26 tháng 8 2016
\(\frac{2}{6}+\frac{2}{12}+...+\frac{2}{x.\left(x+1\right)}=\frac{2011}{2013}\)
\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{2011}{2013}:2\)
\(\frac{1}{2}-\frac{1}{x+1}=\frac{2011}{4026}\)
\(\frac{\left(x+1-2\right)}{2.\left(x+1\right)}=\frac{2011}{4026}\)
AK
26 tháng 7 2018
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)
\(\Leftrightarrow2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2011}{2013}\)
\(\Leftrightarrow2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2011}{2013}\)
\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
\(\Leftrightarrow1-\frac{2}{x+1}=\frac{2011}{2013}\)
\(\Leftrightarrow\frac{2}{x+1}=\frac{2}{2013}\)
\(\Leftrightarrow x+1=2013\)
\(\Leftrightarrow x=2012\)
Vậy \(x=2012\)
HH
26 tháng 7 2018
\(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+.........+\frac{2}{x\left(x+1\right)}=1\frac{2003}{2005}\left(1\right)\)
\(=\frac{2}{1.2}+\frac{2}{2.3}+\frac{2}{3.4}+......+\frac{2}{x\left(x+1\right)}\)
\(=2.\left[\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+........+\frac{1}{x\left(x+1\right)}\right]\)
\(=2.\left[1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.......+\frac{1}{x}-\frac{1}{x+1}\right]\)
\(=2.\left(1-\frac{1}{x+1}\right)\)
\(=2.\left(\frac{x+1}{x+1}-\frac{1}{x+1}\right)\)
\(=2.\frac{x}{x+1}\)
Thay vào ( 1 ) ta có :
\(\frac{2x}{x+1}=\frac{4008}{2005}\Rightarrow\frac{x}{x+1}=\frac{2004}{2005}\)
\(\Rightarrow2005x=2004\left(x+1\right)\Rightarrow2005x=2004.2004\)
\(\Rightarrow2005x=2004x=2004x\Rightarrow x=2004\)
KL : Vậy x = 2004
Đây là bài mẫu của mình bạn dựa theo rồi tự làm nhé
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)
=> \(\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{x\left(x+1\right)}=\frac{2011}{2013}\)
=> \(2.\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2011}{2013}\)
=> \(2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
=> \(2.\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2011}{2013}\)
=> \(2.\frac{1}{2}-2.\frac{1}{x+1}=\frac{2011}{2013}\)
=> \(1-\frac{2}{x+1}=\frac{2011}{2013}\)
=> \(\frac{2}{x+1}=1-\frac{2011}{2013}=\frac{2}{2013}\)
=> x + 1 = 2013
=> x = 2013 - 1 = 2012