A = \(\dfrac{1}{2}+\dfrac{1}{14}+\dfrac{1}{35}+...+\dfrac{1}{152}\)
=> \(\dfrac{3}{2}\)A = \(\dfrac{3}{4}+\dfrac{3}{28}+\dfrac{3}{70}+...+\dfrac{3}{304}\)
= \(\dfrac{3}{1.4}+\dfrac{3}{4.7}+\dfrac{3}{7.10}+...+\dfrac{3}{16.19}\)
= \(1-\dfrac{1}{19}=\dfrac{18}{19}\)
=> A = \(\dfrac{18}{19}:\dfrac{3}{2}=\dfrac{12}{19}\)
\(A=\dfrac{1}{2}+\dfrac{1}{14}+.......+\dfrac{1}{152}\)
\(\Leftrightarrow A=\dfrac{3}{2}\left(\dfrac{3}{4}+\dfrac{3}{28}+...........+\dfrac{3}{304}\right)\)
\(\Leftrightarrow A=\dfrac{3}{2}\left(\dfrac{3}{1.4}+\dfrac{3}{4.7}+.........+\dfrac{3}{16.19}\right)\)
\(\Leftrightarrow A=\dfrac{3}{2}\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+......+\dfrac{1}{16}-\dfrac{1}{19}\right)\)
\(\Leftrightarrow A=\dfrac{3}{2}\left(1-\dfrac{1}{19}\right)\)
\(\Leftrightarrow A=\dfrac{3}{2}.\dfrac{18}{19}=\dfrac{12}{19}\)