\(\dfrac{21}{3\cdot10}+\dfrac{15}{10\cdot15}+\dfrac{27}{15\cdot24}+\dfrac{9}{24\cdot27}\)
\(=3\left(\dfrac{7}{3\cdot10}+\dfrac{5}{10\cdot15}+\dfrac{9}{15\cdot24}+\dfrac{3}{24\cdot27}\right)\)
\(=3\left(\dfrac{1}{3}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{15}+\dfrac{1}{15}-\dfrac{1}{24}+\dfrac{1}{24}-\dfrac{1}{27}\right)\)
\(=3\left(\dfrac{1}{3}-\dfrac{1}{27}\right)=1-\dfrac{1}{9}=\dfrac{8}{9}\)
\(\dfrac{1}{24}+\dfrac{1}{104}+\dfrac{1}{234}+...+\dfrac{1}{924}\)
\(=\dfrac{1}{3\cdot8}+\dfrac{1}{8\cdot13}+\dfrac{1}{13\cdot18}+\dfrac{1}{18\cdot23}+\dfrac{1}{23\cdot28}+\dfrac{1}{28\cdot33}\)
\(=\dfrac{1}{5}\left(\dfrac{5}{3\cdot8}+\dfrac{5}{8\cdot13}+...+\dfrac{5}{28\cdot33}\right)\)
\(=\dfrac{1}{5}\left(\dfrac{1}{3}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{13}+...+\dfrac{1}{28}-\dfrac{1}{33}\right)\)
\(=\dfrac{1}{5}\left(\dfrac{1}{3}-\dfrac{1}{33}\right)=\dfrac{1}{5}\cdot\dfrac{10}{33}=\dfrac{2}{33}\)
\(\left(\dfrac{21}{3\cdot10}+\dfrac{15}{10\cdot15}+\dfrac{27}{15\cdot24}+\dfrac{9}{24\cdot27}\right)\cdot\left(\dfrac{x}{44}-3\right)=\dfrac{1}{24}+\dfrac{1}{104}+...+\dfrac{1}{924}\)
=>\(\dfrac{8}{9}\left(\dfrac{x}{44}-3\right)=\dfrac{2}{33}\)
=>\(\dfrac{x}{44}-3=\dfrac{2}{33}:\dfrac{8}{9}=\dfrac{2}{33}\cdot\dfrac{9}{8}=\dfrac{1}{4}\cdot\dfrac{3}{11}=\dfrac{3}{44}\)
=>\(\dfrac{x}{44}=3+\dfrac{3}{44}=\dfrac{135}{44}\)
=>x=135
