Let a sequence be defined by T_r = r _ i=1 ^r i^3 _ i=1 ^r i for r 1 . If S = _ r=1 ⁴⁰ (-1)^r T_r , then the value of S is equal to

Step-by-step solution

First, simplify the general term T_r using standard summation formulas: T_r = r r^2(r+1)^2 4 r(r+1) 2 = r^2(r+1) 2 The given series is S = _ r=1 ⁴⁰ (-1)^r T_r = -T₁ + T₂ - T₃ + T₄ - - T₃₉ + T₄₀ Grouping the 40 terms into 20 adjacent pairs, we get: S = _ k=1 ²⁰ (T_ 2k - T_ 2k-1 ) Now, calculate T_ 2k - T_ 2k-1 : T_ 2k = (2k)^2(2k+1) 2 = 4k^3 + 2k^2 T_ 2k-1 = (2k-1)^2(2k) 2 = 4k^3 - 4k^2 + k T_ 2k - T_ 2k-1 = (4k^3 + 2k^2) - (4k^3 - 4k^2 + k) = 6k^2 - k Therefore, the sum is: S = _ k=1 ²⁰ (6k^2 - k) = 6 _ k=1 ²⁰ k^2