\(\dfrac{1}{21}+\dfrac{1}{28}+\dfrac{1}{36}+...+\dfrac{1}{x\left(x+1\right):2}=\dfrac{2}{9}\)
<=> \(\dfrac{1}{6.7:2}+\dfrac{1}{7.8:2}+\dfrac{1}{8.9:2}+...+\dfrac{1}{x\left(x+1\right):2}=\dfrac{2}{9}\)
<=> \(\dfrac{2}{6.7}+\dfrac{2}{7.8}+\dfrac{2}{8.9}+...+\dfrac{2}{x\left(x+1\right)}=\dfrac{2}{9}\)
<=> \(2\left(\dfrac{1}{6.7}+\dfrac{1}{7.8}+\dfrac{1}{8.9}+...+\dfrac{1}{x\left(x+1\right)}\right)=\dfrac{2}{9}\)
<=> \(2\left(\dfrac{1}{6}-\dfrac{1}{x+1}\right)=\dfrac{2}{9}\)
<=> \(\dfrac{1}{6}-\dfrac{1}{x+1}=\dfrac{1}{9}\)
<=> \(\dfrac{1}{x+1}=\dfrac{1}{18}\)
<=> x + 1 = 18
<=> x = 17
\(\dfrac{1}{21}+\dfrac{1}{28}+\dfrac{1}{36}+...+\dfrac{1}{x.\left(x+1\right):2}=\dfrac{2}{9}\)
\(\Leftrightarrow\dfrac{1}{2}\left(\dfrac{1}{21}+\dfrac{1}{28}+\dfrac{1}{36}+...+\dfrac{1}{x\left(x+1\right):2}\right)=\dfrac{2}{9}.2=\dfrac{4}{9}\)\(\Leftrightarrow\dfrac{1}{42}+\dfrac{1}{56}+\dfrac{1}{72}+...+\dfrac{1}{x\left(x+1\right)}=\dfrac{4}{9}\)
\(\Leftrightarrow\dfrac{1}{6.7}+\dfrac{1}{7.8}+\dfrac{1}{8.9}+...+\dfrac{1}{x}+\dfrac{1}{x+1}=\dfrac{4}{9}\)\(\Leftrightarrow\dfrac{1}{6}+\dfrac{1}{7}-\dfrac{1}{7}+\dfrac{1}{8}-\dfrac{1}{8}+...+\dfrac{1}{x}+\dfrac{1}{x+1}=\dfrac{4}{9}\)\(\Leftrightarrow\dfrac{1}{6}-\dfrac{1}{x+1}=\dfrac{4}{9}\)
\(\Leftrightarrow\dfrac{1}{x+1}=\dfrac{4}{9}-\dfrac{1}{6}=\dfrac{5}{8}\)
\(\Leftrightarrow\left(1.8\right)=5\left(x+1\right)\)
\(\Leftrightarrow8=5x+5\)
\(\Leftrightarrow5x=8-3=5\)
\(\Leftrightarrow x=5:5\)
\(\Leftrightarrow x=1\)