\(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{x.\left(x+1\right)}=\frac{2}{9}\)
\(\frac{2}{42}+\frac{2}{56}+\frac{2}{72}+...+\frac{2}{x.\left(x+1\right)}=\frac{2}{9}\)
\(\frac{2}{6.7}+\frac{2}{7.8}+\frac{2}{8.9}+...+\frac{2}{x.\left(x+1\right)}=\frac{2}{9}\)
\(2.\left(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{x.\left(x+1\right)}\right)=\frac{2}{9}\)
\(2.\left(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...+\frac{1}{x}-\frac{1}{x-1}\right)=\frac{2}{9}\)
\(2.\left(\frac{1}{6}-\frac{1}{x-1}\right)=\frac{2}{9}\)
\(\frac{1}{6}-\frac{1}{x-1}=\frac{2}{9}:2\)
\(\frac{1}{6}-\frac{1}{x-1}=\frac{1}{9}\)
\(\frac{1}{x-1}=\frac{1}{6}-\frac{1}{9}\)
\(\frac{1}{x-1}=\frac{1}{18}\)
\(\Rightarrow x-1=18\)
\(\Rightarrow x=18+1\)
\(\Rightarrow x=19\)
= 2/42 + 2/56+2/72+................+2/x.(x+1)=2/9
=\(\frac{2}{6.7}\)+\(\frac{2}{7.8}\)+\(\frac{2}{8.9}\)+......+\(\frac{2}{x.\left(x+1\right)}\)=2/9
=2.( \(\frac{1}{6}\)-\(\frac{1}{7}\)+\(\frac{1}{7}\)-\(\frac{1}{8}\)+.......+\(\frac{1}{x}\)-\(\frac{1}{x+1}\)
=2.(1/6 -\(\frac{1}{x+1}\))=2/9
=1/6 -\(\frac{1}{x+1}\)=2/9:2=1/9
=1/6-1/9=\(\frac{1}{x+1}\)=3/54=1/18
=> x= 18-1 =17
