\(\Leftrightarrow\frac{1}{2}\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+...+\frac{2}{\left(5x+1\right)\left(5x+3\right)}\right)=\frac{11}{23}\)
\(\Leftrightarrow\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{5x+1}-\frac{1}{5x+3}\right)=\frac{11}{23}\)
\(\Leftrightarrow1-\frac{1}{5x+3}=\frac{22}{23}\)
\(\Leftrightarrow\frac{1}{5x+3}=\frac{1}{23}\)
\(\Leftrightarrow5x+3=23\Leftrightarrow x=4\) ( TM )
\(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{\left(5x+1\right).\left(5x+3\right)}=\frac{11}{23}\)
\(\Rightarrow\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{\left(5x+1\right)\left(5x+3\right)}\right)=\frac{11}{23}\)
\(\Rightarrow\frac{1}{2}\left(1-\frac{1}{3}+...+\frac{1}{\left(5x+1\right)}-\frac{1}{\left(5x+3\right)}\right)=\frac{11}{23}\)
\(\Rightarrow1-\frac{1}{\left(5x+3\right)}=\frac{11}{23}:\frac{1}{2}\)
\(\Rightarrow\frac{1}{5x+3}=\frac{1}{23}\)
\(\Rightarrow5x+3=23\)
\(\Rightarrow5x=23-3\)
\(\Rightarrow x=20:5\)
\(\Rightarrow x=4\)
