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Question Number 212066 by Spillover last updated on 28/Sep/24

Answered by Rasheed.Sindhi last updated on 29/Sep/24

$$\frac{2021!+2020!}{2021!-2020!}\cdot \frac{2020!+2019!}{2020!-2019!}\cdot ...\frac{3!+2!}{3!-2!}\cdot \frac{2!+1!}{2!-1!}$$$$\frac{\{(x+1)!+x!\}}{\{(x+1)!-x!\}}=\frac{x!\{(x+1)+1\}}{x!\{(x+1)-1\}}=\frac{x+2}{x}$$$$\frac{2020+2}{2020}\cdot \frac{2019+2}{2019}\cdot ...\frac{2+2}{2}\cdot \frac{1+2}{1}$$$$\frac{(2020+2)(2019+2)(2018+2)...(2+2)(1+2)}{2020.2019.2018....2.1}$$$$=\frac{2022.2021.2020....4.3}{2020!}$$$$=\frac{2022.2021.2020....4.3.(2.1)}{2020!(2.1)}$$$$=\frac{2022.2021}{2}=2043231$$

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