Ta có: \(1+\dfrac{1}{n.\left(n+2\right)}=\dfrac{n^2+2n+1}{n\left(n+2\right)}=\dfrac{\left(n+1\right)^2}{n\left(n+1\right)}\)
Áp dụng:
\(A=\dfrac{1}{2}\left[\left(1+\dfrac{1}{1.3}\right)\left(1+\dfrac{1}{2.4}\right)...\left(1+\dfrac{1}{2022.2024}\right)\right]\)
\(=\dfrac{1}{2}\left[\dfrac{2^2.3^2...2023^2}{1.3.2.4...2022.2024}\right]=\dfrac{1}{2}\times\dfrac{2.3...2023}{1.2...2022}\times\dfrac{2.3...2023}{3.4...2024}=\dfrac{1}{2}\times\dfrac{2023}{1}\times\dfrac{2}{2024}\)
\(=\dfrac{2023}{2024}\)
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