Ta có: \(x-\frac{20}{11\cdot13}-\frac{20}{13\cdot15}-...-\frac{20}{53\cdot55}=\frac{3}{11}\)

\(\Leftrightarrow x-10\cdot\left(\frac{2}{11\cdot13}+\frac{2}{13\cdot15}+...+\frac{2}{53\cdot55}\right)=\frac{3}{11}\)

\(\Leftrightarrow x-10\cdot\left(\frac{1}{11}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}+...+\frac{1}{53}-\frac{1}{55}\right)=\frac{3}{11}\)

\(\Leftrightarrow x-10\cdot\left(\frac{1}{11}-\frac{1}{55}\right)=\frac{3}{11}\)

\(\Leftrightarrow x-10\cdot\frac{4}{55}=\frac{3}{11}\)

\(\Leftrightarrow x-\frac{8}{11}=\frac{3}{11}\)

\(\Leftrightarrow x=\frac{3}{11}+\frac{8}{11}\)

\(\Leftrightarrow x=1\)

Vậy \(x=1\)thỏa mãn đề. 

Vi ơi, bài đội tuyển hả?

\(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+...+\frac{2}{x\left(x+1\right)}=\frac{2}{9}\)

\(\frac{1}{2}\left(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{x\left(x+1\right)}\right)=\frac{2}{9}\cdot\frac{1}{2}\)

\(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{1}{x\left(x+1\right)}=\frac{1}{9}\)

\(\frac{1}{6\cdot7}+\frac{1}{7\cdot8}+\frac{1}{8\cdot9}+...+\frac{1}{x\left(x+1\right)}=\frac{1}{9}\)

\(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{1}{9}\)

\(\frac{1}{6}-\frac{1}{x+1}=\frac{1}{9}\)

\(\frac{1}{x+1}=\frac{1}{6}-\frac{1}{9}\)

\(\frac{1}{x+1}=\frac{1}{18}\)

\(\Rightarrow x+1=18\)

\(x=18-1\)

\(x=17\)

sửa đề số cuối vế trái là \(\frac{1}{x\left(x+1\right)}\)

Đặt A là vế trái

\(\frac{1}{2}A=\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{1}{x\left(x+1\right)}\)

\(=\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x\left(x+1\right)}\)

\(=\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...-\frac{1}{x}+\frac{1}{x}-\frac{1}{x+1}\)

\(=\frac{1}{6}-\frac{1}{x+1}\)

\(\Rightarrow A=\frac{1}{3}-\frac{2}{x+1}=\frac{2}{9}\)

\(\frac{2}{x+1}=\frac{1}{3}-\frac{2}{9}=\frac{1}{9}=\frac{2}{18}\)

\(\Rightarrow x+1=18\Rightarrow x=17\)

Vậy x=17

bạn Nga Nguyễn làm sai rồi , đề bài này hoàn toàn đúng , vì :

\(\frac{2}{x\left(x+1\right)}\cdot\frac{1}{2}=\frac{2\cdot1}{x\left(x+1\right)\cdot2}=\frac{1}{x\left(x+1\right)}\)

còn nếu theo cách của bạn , sẽ ra :

\(\frac{1}{x\left(x+1\right)}\cdot\frac{1}{2}=\frac{1}{x\left(x+1\right)\cdot2}\)

sai nhưng kết quả đúng ? đkp

* ĐK: \(x\ne0\)

Đề ra ...<=> \(\frac{2}{42}+\frac{2}{56}+\frac{2}{72}+...+\frac{1}{x\left(x+1\right)}=\frac{2}{9}\)

<=> \(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{2}{x\left(x+1\right)}=\frac{1}{9}\)

<=> \(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x\left(x+1\right)}+\frac{1}{x\left(x+1\right)}=\frac{1}{9}\)

<=>\(\frac{1}{6}-\frac{1}{x+1}+\frac{1}{x\left(x+1\right)}=\frac{1}{9}\)

<=>\(\frac{1}{x+1}\left(1-\frac{1}{x}\right)=\frac{1}{6}-\frac{1}{9}\)

<=> \(\frac{x-1}{x\left(x+1\right)}=\frac{1}{36}\)

<=> \(\frac{x-1}{x\left(x-1\right)}=\frac{x-1}{36.\left(x-1\right)}\)

=> x(x-1) = 36. (x-1) => x =36

\(\frac{2}{2}.\left(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{x+\left(x+1\right)}\right)=\frac{2}{9}\)

\(2\left(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{1}{x.\left(x+1\right)}\right)=\frac{2}{9}\)

\(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x.\left(x+1\right)}=\frac{1}{9}\)

\(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{1}{9}\)

\(\frac{1}{6}-\frac{1}{x+1}=\frac{1}{9}\)

\(\frac{1}{x+1}=\frac{1}{6}-\frac{1}{9}\)

\(\frac{1}{x+1}=\frac{1}{18}\)

x+1=18

x=18-1

x=17

A) \(\frac{1}{2}\cdot\left(\frac{2}{9}+\frac{3}{7}-\frac{5}{27}\right)\) 

\(=\frac{1}{2}\cdot\frac{1}{2}\)

\(=\frac{1}{4}\)

B)   \(\left(\frac{-5}{28}+1.75+\frac{8}{35}\right):\left(-3\frac{9}{20}\right)\)

\(=\left(\frac{-5}{28}+\frac{7}{4}+\frac{8}{35}\right):\frac{-69}{20}\)

\(=\frac{14}{5}:\frac{-69}{20}\)

\(=\frac{-56}{69}\)

Sửa đề : \(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{x\left(x+1\right)}=\frac{2}{9}\)

\(\Leftrightarrow\)\(\frac{2}{42}+\frac{2}{56}+\frac{2}{72}+...+\frac{2}{x\left(x+1\right)}=\frac{2}{9}\)

\(\Leftrightarrow\)\(2\left(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2}{9}\)

\(\Leftrightarrow\)\(2\left(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2}{9}\)

\(\Leftrightarrow\)\(2\left(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2}{9}\)

\(\Leftrightarrow\)\(2\left(\frac{1}{6}-\frac{1}{x+1}\right)=\frac{2}{9}\)

\(\Leftrightarrow\)\(\frac{1}{6}-\frac{1}{x+1}=\frac{2}{9}.\frac{1}{2}\)

\(\Leftrightarrow\)\(\frac{1}{6}-\frac{1}{x+1}=\frac{1}{9}\)

\(\Leftrightarrow\)\(\frac{1}{x+1}=\frac{1}{6}-\frac{1}{9}\)

\(\Leftrightarrow\)\(\frac{1}{x+1}=\frac{1}{18}\)

\(\Leftrightarrow\)\(x+1=18\)

\(\Leftrightarrow\)\(x=18-1\)

\(\Leftrightarrow\)\(x=17\)

Vậy \(x=17\)

Chúc bạn học tốt ~ 

\(x-\frac{20}{11.13}-\frac{20}{13.15}-\frac{20}{15.17}-...-\frac{20}{53.55}=\frac{3}{11}\)

\(\Leftrightarrow\)\(x+10\left(\frac{2}{11.13}+\frac{2}{13.15}+\frac{2}{15.17}+...+\frac{2}{53.55}\right)=\frac{3}{11}\)

\(\Leftrightarrow\)\(x+10\left(\frac{1}{11}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}+\frac{1}{15}-\frac{1}{17}+...+\frac{1}{53}-\frac{1}{55}\right)=\frac{3}{11}\)

\(\Leftrightarrow\)\(x+10\left(\frac{1}{11}-\frac{1}{55}\right)=\frac{3}{11}\)

\(\Leftrightarrow\)\(x+10.\frac{4}{55}=\frac{3}{11}\)

\(\Leftrightarrow\)\(x+\frac{40}{55}=\frac{3}{11}\)

\(\Leftrightarrow\)\(x=\frac{3}{11}-\frac{40}{55}\)

\(\Leftrightarrow\)\(x=\frac{-5}{11}\)

Vậy \(x=\frac{-5}{11}\)

Chúc bạn học tốt ~ 

\(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{1}{x\left(x+1\right)}=\frac{2}{9}\)

\(\Rightarrow\frac{2}{42}+\frac{2}{56}+\frac{2}{72}+...+\frac{1}{x\left(x+1\right)}=\frac{2}{9}\)

\(\Rightarrow\frac{2}{6\cdot7}+\frac{2}{7\cdot8}+\frac{2}{8\cdot9}+...+\frac{1}{x\left(x+1\right)}=\frac{2}{9}\)

\(\Rightarrow2\left(\frac{1}{6\cdot7}+\frac{1}{7\cdot8}+\frac{1}{8\cdot9}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2}{9}\)

\(\Rightarrow2\left(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2}{9}\)

\(\Rightarrow2\left(\frac{1}{6}-\frac{1}{x+1}\right)=\frac{2}{9}\)

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

\(\Rightarrow\frac{2}{x+1}=\frac{1}{9}\)

=> 2.9 = x + 1

=> x + 1 = 18

=> x = 17

Nhân vế trái với 1/2 là sẽ ra nhé bạn !!

Nó sẽ ra kiểu : 1/6.7+1/7.8+.....+1/x.(x+1)=2/9

Nhân cả 1 vế với 1/2 mới đúng

mk nhầm nhân cả 2 vế với 1/2