\(=\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{2021}-\dfrac{1}{2024}=\dfrac{1}{4}-\dfrac{1}{2024}=\dfrac{505}{2024}\)
\(\dfrac{3}{4.7}+\dfrac{3}{7.10}+\dfrac{3}{10.13}+...+\dfrac{1}{2021.2024}\)
=\(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{14}+...+\dfrac{1}{2021}-\)\(\dfrac{1}{2024}\)
=\(\dfrac{1}{4}-\dfrac{1}{2024}\)
=\(\dfrac{505}{2024}\)
\(\dfrac{3}{4.7}+\dfrac{3}{7.10}+\dfrac{3}{10.13}+...+\dfrac{3}{2021.2024}\)
\(=\dfrac{3}{3}.\left(\dfrac{3}{4.7}+\dfrac{3}{7.10}+\dfrac{3}{10.13}+...+\dfrac{3}{2021.2024}\right)\)
\(=1.\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+\dfrac{1}{10}-\dfrac{1}{13}+...+\dfrac{1}{2021}-\dfrac{1}{2024}\right)\)
\(=1.\left(\dfrac{1}{4}-\dfrac{1}{2024}\right)\)
\(=1.\left(\dfrac{506-1}{2024}\right)\)
\(=1.\dfrac{505}{2024}\)
\(=\dfrac{505}{2024}\)
