So sánh 2 biểu thức sau A = 1 + 2022^2022 / 1 + 2022^2023 và B = 1 + 2022^2023/ 1 + 2022^2024 .
Giải thích
Ta có: \(2022A = \frac{{2022\left( {1 + {{2022}^{2022}}} \right)}}{{1 + {{2022}^{2023}}}} = \frac{{2022 + {{2022}^{2023}}}}{{1 + {{2022}^{2023}}}} = 1 + \frac{{2021}}{{1 + {{2022}^{2023}}}}\);
\(2022B = \frac{{2022\left( {1 + {{2022}^{2023}}} \right)}}{{1 + {{2022}^{2024}}}} = \frac{{2022 + {{2022}^{2024}}}}{{1 + {{2022}^{2024}}}} = 1 + \frac{{2021}}{{1 + {{2022}^{2024}}}}\).
Vì \(\frac{1}{{1 + {{2022}^{2023}}}} > \frac{1}{{1 + {{2022}^{2024}}}}\) nên \(\frac{{2021}}{{1 + {{2022}^{2023}}}} > \frac{{2021}}{{1 + {{2022}^{2024}}}}\)
Suy ra \(1 + \frac{{2021}}{{1 + {{2022}^{2023}}}} > 1 + \frac{{2021}}{{1 + {{2022}^{2024}}}}\) hay \(2022A < 2022B\).
Vậy \(A < B\).