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(ii) 
1^(3)-2^(3)+3^(3)-cdots+19^(3)-20^(3)

(ii) 1323+33+193203 1^{3}-2^{3}+3^{3}-\cdots+19^{3}-20^{3}

Full solution

Q. (ii) 1323+33+193203 1^{3}-2^{3}+3^{3}-\cdots+19^{3}-20^{3}
  1. Calculate Cubes: Calculate the cubes of each number from 11 to 2020 and alternate the signs starting with positive for 131^3.$13=1\$1^3 = 1, 23=82^3 = 8, 33=273^3 = 27, ..., 193=685919^3 = 6859, 203=800020^3 = 8000).
  2. Sum Series with Signs: Sum the series with alternating signs: 18+2764++685980001 - 8 + 27 - 64 + \ldots + 6859 - 8000. The series alternates between adding and subtracting the cubes of consecutive integers.
  3. Use Sum Formula: Use the formula for the sum of cubes of the first nn natural numbers: (n(n+1)2)2\left(\frac{n(n+1)}{2}\right)^2. For odd nn, the sum is positive, and for even nn, the sum is negative. Sum = (1323)+(3343)++(193203)(1^3 - 2^3) + (3^3 - 4^3) + \ldots + (19^3 - 20^3).

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