1. **Problem statement:** Calculate the sums: $$M = 1 + (-2) + 3 + (-4) + \cdots + 2001 + (-2002)$$ $$A = 7 - 8 + 9 - 10 + 11 - 12 + \cdots + 2009 - 2010$$ $$C = 2 - 4 + 6 - 8 + \cdots + 1998 - 2000$$ $$D = 2 - 4 - 6 + 8 + 10 - 12 - 14 + 16 + \cdots + 1994 - 1996 - 1998 + 2000 + 2009$$ $$d = (-8)^2 \cdot 19 + 19 \cdot (-6)^2$$ 2. **Sum M:** - The terms alternate between positive odd numbers and negative even numbers. - Number of terms: from 1 to 2002, total 2002 terms. - Odd terms count: 1001 (all positive odd numbers from 1 to 2001). - Even terms count: 1001 (all negative even numbers from -2 to -2002). Sum of positive odd numbers: $$\sum_{k=1}^{1001} (2k-1) = 1001^2 = 1002001$$ Sum of negative even numbers: $$-\sum_{k=1}^{1001} 2k = -2 \cdot \frac{1001 \cdot 1002}{2} = -1001 \cdot 1002 = -1003002$$ Therefore, $$M = 1002001 - 1003002 = -1001$$ 3. **Sum A:** - Terms start at 7 and alternate signs every two terms: +7 -8 +9 -10 +11 -12 ... - Group terms in pairs: (7 - 8), (9 - 10), (11 - 12), ... - Each pair sums to -1. - Number of pairs: from 7 to 2010, total terms = 2010 - 7 + 1 = 2004 terms. - Number of pairs = 2004 / 2 = 1002 pairs. Sum: $$A = 1002 \times (-1) = -1002$$ 4. **Sum C:** - Even numbers from 2 to 2000 with alternating signs starting positive: 2 - 4 + 6 - 8 + ... - Number of terms: 1000 (since 2000/2 = 1000). - Group terms in pairs: (2 - 4), (6 - 8), ... each pair sums to -2. - Number of pairs: 1000 / 2 = 500 pairs. Sum: $$C = 500 \times (-2) = -1000$$ 5. **Sum D:** - Pattern: 2 - 4 - 6 + 8 + 10 - 12 - 14 + 16 + ... + 1994 - 1996 - 1998 + 2000 + 2009 - The pattern repeats every 4 terms: +, -, -, + - Group terms in blocks of 4: Block sum = (2) + (-4) + (-6) + (8) = 0 - Number of full blocks: 1996 / 4 = 499 blocks (since 1996 is divisible by 4) - Sum of these blocks = 499 * 0 = 0 - Remaining terms: 1998, 2000, 2009 - Signs for these terms follow the pattern: 1998: - 2000: + 2009: + Sum of remaining terms: $$-1998 + 2000 + 2009 = 11$$ Therefore, $$D = 0 + 11 = 11$$ 6. **Expression d:** Calculate: $$(-8)^2 \cdot 19 + 19 \cdot (-6)^2$$ Calculate squares: $$(-8)^2 = 64, \quad (-6)^2 = 36$$ Calculate products: $$64 \cdot 19 = 1216, \quad 19 \cdot 36 = 684$$ Sum: $$d = 1216 + 684 = 1900$$ **Final answers:** $$M = -1001$$ $$A = -1002$$ $$C = -1000$$ $$D = 11$$ $$d = 1900$$