\(A=\dfrac{4}{1.2}+\dfrac{4}{2.3}+...+\dfrac{4}{1010.1011}\\ =4\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{1010.1011}\right)\\ =4\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{1010}-\dfrac{1}{1011}\right)\\ =4\left(1-\dfrac{1}{1011}\right)\\ =4.\dfrac{1010}{1011}\\ =\dfrac{4040}{1011}\)
\(A=4.\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+.....+\dfrac{1}{1010.1011}\right)\)
\(A=4.\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+....+\dfrac{1}{1010}-\dfrac{1}{1011}\right)\)
\(A=4.\left(1-\dfrac{1}{1011}\right)\)
\(A=4.\dfrac{1010}{1011}\)
\(A=\dfrac{4040}{1010}\)
\(A=4\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{1010.1011}\right)\)
\(A=4\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{1010}-\dfrac{1}{1011}\right)\)
\(A=4\left(1-\dfrac{1}{1010}\right)=4\left(\dfrac{1011}{1011}-\dfrac{1}{1011}\right)=\dfrac{4040}{1011}\)