\(\frac{1}{20}+\frac{1}{44}+\frac{1}{77}+...+\frac{2}{x\left(x+3\right)}=\frac{101}{770}\)
\(\Rightarrow\)\(\frac{3}{2}.\left(\frac{1}{20}+\frac{1}{44}+\frac{1}{77}+...+\frac{2}{x\left(x+3\right)}\right)=\frac{101}{770}\).
\(\Rightarrow\)\(\frac{3}{40}+\frac{3}{88}+\frac{3}{154}+...+\frac{3}{x\left(x-3\right)}=\frac{303}{1540}\)
\(\Rightarrow\)\(\frac{3}{5.8}+\frac{3}{8.11}+\frac{3}{11.14}+...+\frac{3}{x\left(x+3\right)}=\frac{303}{1540}\)
\(\Rightarrow\)\(\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+\frac{1}{11}-\frac{1}{14}+...+\frac{1}{x-1}\)
\(\Rightarrow\)\(\frac{1}{5}-\frac{1}{x+3}=\frac{303}{1540}\)
\(\Rightarrow\)\(\frac{1}{x+3}=\frac{1}{5}-\frac{303}{1540}\)
\(\Rightarrow\)\(\frac{1}{x+3}=\frac{308}{1540}-\frac{303}{1540}\)
\(\Rightarrow\)\(\frac{1}{x+3}=\frac{5}{1540}\)
\(\Rightarrow\)\(\frac{1}{x+3}=\)\(\frac{1}{308}\)
\(\Rightarrow\)\(x+3=308\)
\(\Rightarrow\)\(x=308-3\)
\(\Rightarrow\)\(x=305\)
