\(\dfrac{7}{4}.\left(\dfrac{33}{12}+\dfrac{3333}{2020}+\dfrac{333333}{303030}+\dfrac{33333333}{42424242}\right)\)
\(=\dfrac{7}{4}.\left(\dfrac{33}{12}+\dfrac{33}{20}+\dfrac{33}{30}+\dfrac{33}{42}\right)\)
\(=\dfrac{7}{4}.33\left(\dfrac{1}{12}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}\right)\)
\(=\dfrac{7}{4}.33\left(\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{6.7}\right)\)
\(=\dfrac{7}{4}.33\left(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\right)\)
\(=\dfrac{7}{4}.33\left(\dfrac{1}{3}-\dfrac{1}{7}\right)\)
\(=\dfrac{7}{4}.33\left(\dfrac{7}{21}-\dfrac{3}{21}\right)\)
\(=\dfrac{7}{4}.33.\dfrac{4}{21}\)
\(=\dfrac{231}{4}.\dfrac{4}{21}\)
\(=\dfrac{231}{21}=11\)
