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nice telescoping series

Source: 2022 AMC 10B #9

November 17, 2022
AMCAMC 102022 AMC2022 AMC 10B

Problem Statement

The sum 12!+23!+34!+⋯+20212022!\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!} can be expressed as a−1b!a-\frac{1}{b!}, where aa and bb are positive integers. What is a+ba+b?
<spanclass=′latex−bold′>(A)</span> 2020<spanclass=′latex−bold′>(B)</span> 2021<spanclass=′latex−bold′>(C)</span> 2022<spanclass=′latex−bold′>(D)</span> 2023<spanclass=′latex−bold′>(E)</span> 2024 <span class='latex-bold'>(A)</span>\ 2020 \qquad<span class='latex-bold'>(B)</span>\ 2021 \qquad<span class='latex-bold'>(C)</span>\ 2022 \qquad<span class='latex-bold'>(D)</span>\ 2023 \qquad<span class='latex-bold'>(E)</span>\ 2024
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