\(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{n\left(n+1\right)}=\frac{2}{9}\)
\(\frac{1}{2}\left(\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+...+\frac{2}{n\left(n+1\right)}\right)=\frac{2}{9}.\frac{1}{2}\)
\(\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+....+\frac{1}{n\left(n+1\right)}=\frac{1}{9}\)
\(\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+...+\frac{1}{n\left(n+1\right)}=\frac{1}{9}\)
\(\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+...+\frac{1}{n}-\frac{1}{n+1}=\frac{1}{9}\)
\(\frac{1}{6}-\frac{1}{n+1}=\frac{1}{9}\)
\(\frac{1}{n+1}=\frac{1}{6}-\frac{1}{9}\)
\(\frac{1}{n+1}=\frac{1}{18}\)
\(\Rightarrow n+1=18\)
\(\Rightarrow n=17\)
