Bài 1 : 

\(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{\left(2x+1\right)\left(2x+3\right)}=\frac{9}{19}\)

\(\Leftrightarrow1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2x+1}-\frac{1}{2x+3}=\frac{9}{19}\)

\(\Leftrightarrow1-\frac{1}{2x+3}=\frac{9}{19}\)

\(\Leftrightarrow\frac{1}{2x+3}=1-\frac{9}{19}\)

\(\Leftrightarrow\frac{1}{2x+3}=\frac{10}{19}\)

\(\Leftrightarrow10.\left(2x+3\right)=19\Leftrightarrow2x+3=\frac{19}{10}\)

\(\Leftrightarrow2x=\frac{19}{10}-3\Leftrightarrow2x=-\frac{11}{10}\)

\(\Leftrightarrow x=-\frac{11}{20}=-0,55\)

Bài 2 : 

\(\frac{2}{2.4}+\frac{2}{4.6}+\frac{2}{6.8}+...+\frac{2}{2016.2018}\)

\(=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+....+\frac{1}{2016}-\frac{1}{2018}\)

\(=\frac{1}{2}-\frac{1}{2018}=\frac{504}{1009}\)

số thập phân ghi làm sao

tớ làm được rồi

\(A=\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{2012.2014}\)

\(\Leftrightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{2012}-\frac{1}{2014}\right)\)

\(\Leftrightarrow A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{2014}\right)\)

\(\Leftrightarrow A=\frac{1}{2}\cdot\frac{503}{1007}\)

\(\Leftrightarrow A=\frac{503}{2014}\)

= 1/2[1/2 - 1/4+1/4-1/6 + 1/6-1/8+...+ 1/2012-1/2014]

= 1/2[1/2-1/2014]

= 1/2 * 503/1007

= 503/2014

\(B=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{96}-\frac{1}{98}+\frac{1}{98}-\frac{1}{100}\right)\)

\(B=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{100}\right)=\frac{1}{2}.\frac{49}{100}=\frac{49}{200}\)

Bn Nguyễn Tuấn Minh làm đúng rồi đó bạn

Gán A=2 ; B=0 Nhập công thức :   B=1/A(A+2) : A=A+2 : C=C+B

Ta có : \(\frac{1}{10.9}-\frac{1}{9.8}-.....-\frac{1}{2.1}\)

\(=\frac{1}{90}-\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.....+\frac{1}{9.8}\right)\)

\(=\frac{1}{90}-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{8}-\frac{1}{9}\right)\)

\(=\frac{1}{90}-\left(1-\frac{1}{9}\right)\)

\(=\frac{1}{90}-\frac{8}{9}=\frac{-79}{90}\)

\(A=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{6\cdot7}+\frac{1}{7\cdot8}\)

\(A=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}\)

\(A=\frac{1}{2}-\frac{1}{8}=\frac{3}{8}\)

\(A=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}\\ \)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{7}-\frac{1}{8}\)

\(=\frac{1}{2}-\frac{1}{8}\)

\(=\frac{4}{8}-\frac{1}{8}\\ =\frac{3}{8}\)

Chúc bn học thiệt giỏi nhé!

xét          \(VT=\frac{2}{2}\left(\frac{1}{2.4}+\frac{1}{4.6}+......+\frac{1}{2n.\left(2n+2\right)}\right)\)     (1)

\(=\frac{1}{2}\left(\frac{2}{2.4}+\frac{2}{4.6}+.......+\frac{2}{2n\left(2n+2\right)}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+.......+\frac{1}{2n}-\frac{1}{2n+2}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)=\frac{1}{4}-\frac{1}{2\left(2n+2\right)}\)

\(=\frac{1}{4}-\frac{1}{4n+4}\)

mà theo bài ra   (1) = \(\frac{502}{2009}\)

<=>\(\frac{1}{4}-\frac{1}{4n+4}=\frac{502}{2009}\)

<=>\(\frac{1}{4n+4}=\frac{1}{4}-\frac{502}{2009}\)

<=>\(\frac{1}{4n+4}=\frac{1}{8036}\)

<=> 4n+4=8036

<=> 4n=8032

<=> n=2008

=) \(\frac{1}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{2n\left(2n+2\right)}\right)=\frac{502}{2009}\)
=) \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2n}-\frac{1}{2n+2}\right)=\frac{502}{2009}\)
=) \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)=\frac{502}{2009}\)
=) \(\frac{1}{2}-\frac{1}{2n+2}=\frac{502}{2009}:\frac{1}{2}=\frac{1018}{2009}\)
=) \(\frac{1}{2n+2}=\frac{1}{2}-\frac{1018}{2009}=\frac{-27}{4018}\)
=) \(\frac{-1}{-\left(2n+2\right)}=\frac{-27}{4018}\)
=) \(\frac{-27}{27.-\left(2n+2\right)}=\frac{-27}{4018}\)
=) \(27.-\left(2n+2\right)=4018\)
=) \(-\left(2n+2\right)=4018:27=\frac{4018}{27}\)
=) \(2n+2=\frac{-4018}{27}\)
=) \(2n=\frac{-4018}{27}-2=\frac{-4072}{27}\)
=) \(n=\frac{-4072}{27}:2=\frac{-2036}{27}\)
\(\)