1. Stating the problem: Calculate the value of the expression $$\frac{1+2-3-4+5+6-7-8+\cdots+2021+2022-2023-2024}{1-2+3-4+5-6+7-8+\cdots+2023-2024+2025}$$. 2. Analyze the numerator pattern: The numerator groups every four terms as $(1+2-3-4)$, $(5+6-7-8)$, and so on, up to $(2021+2022-2023-2024)$. 3. Calculate one group sum in numerator: $$1+2-3-4 = (1+2)-(3+4) = 3-7 = -4$$ Each group of four terms sums to $-4$. 4. Count the number of groups in numerator: Since terms go from 1 to 2024, total terms = 2024. Number of groups = $\frac{2024}{4} = 506$. 5. Sum numerator: $$506 \times (-4) = -2024$$ 6. Analyze the denominator pattern: The denominator is an alternating sum of consecutive integers starting with $1$, alternating signs every term: $$1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \cdots + 2023 - 2024 + 2025$$ 7. Group denominator terms in pairs (except last term): $$(1-2) + (3-4) + (5-6) + \cdots + (2023-2024) + 2025$$ Each pair sums to $-1$. 8. Count pairs in denominator: Number of pairs = $\frac{2024}{2} = 1012$. 9. Sum pairs: $$1012 \times (-1) = -1012$$ 10. Add last term: $$-1012 + 2025 = 1013$$ 11. Final expression value: $$\frac{-2024}{1013}$$ 12. Match with options: This corresponds to option D. Final answer: **-2024 / 1013**