问题538: 2×2/1×3+4×4/3×5+6×6/5×7+8×8/7×9+⋯+2024×2024/2023×2025

$$ \begin{aligned} & \frac{2 \times 2}{1 \times 3}+\frac{4 \times 4}{3 \times 5}+\frac{6 \times 6}{5 \times 7}+\frac{8 \times 8}{7 \times 9}+\cdots+\frac{2024 \times 2024}{2023 \times 2025} \\ = & \frac{1 \times 3+1}{1 \times 3}+\frac{3 \times 5+1}{3 \times 5}+\frac{5 \times 7+1}{5 \times 7}+\frac{7 \times 9+1}{7 \times 9}+\cdots+\frac{2023 \times 2025+1}{2023 \times 2025} \\ = & 1+\frac{1}{1 \times 3}+1+\frac{1}{3 \times 5}+1+\frac{1}{5 \times 7}+1+\frac{1}{7 \times 9}+\cdots+1+\frac{1}{2023 \times 2025} \\ = & 1012+\frac{1}{2} \times\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\cdots+\frac{1}{2023}-\frac{1}{2025}\right) \\ = & 1012+\frac{1}{2} \times\left(1-\frac{1}{2025}\right) \\ = & 1012 \frac{1012}{2025}\end{aligned} $$

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