Question

Answer 1

\(=\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2021}-\dfrac{1}{2022}\\ =1+\left(-\dfrac{1}{3}+\dfrac{1}{3}\right)+\left(-\dfrac{1}{4}+\dfrac{1}{4}\right)+...+\left(-\dfrac{1}{2021}+\dfrac{1}{2021}\right)-\dfrac{1}{2022}\\ =1-\dfrac{1}{2022}=\dfrac{2021}{2022}\)

Answer 2

\(\dfrac{1}{2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2021.2022}\)

\(=\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2021}-\dfrac{1}{2022}\)

\(=\dfrac{1}{2}+\dfrac{1}{2}+\left(-\dfrac{1}{3}+\dfrac{1}{3}\right)+...+\left(-\dfrac{1}{2021}+\dfrac{1}{2021}\right)-\dfrac{1}{2022}\)

\(=1-\dfrac{1}{2022}\)

\(=\dfrac{2022}{2022}-\dfrac{1}{2022}\)

\(=\dfrac{2021}{2022}\)