PIMSO 2022 - Mathematics - Grade 7 / Secondary 1 - National Round Question 13

Simplify: $$2023\left[\frac1{2\times1}+\frac2{3\times4}+\frac3{4\times9}+\frac4{5\times16}+\cdots+\frac{2021}{2022\times2021^2}+\frac{2022}{2023\times2022^2}\right].$$