Let i = - 1 . If - 1 + i 3 21 1 - i 24 + 1 + i 3 21 1 + i 24 = k , and n = k be the greatest integral part of k . Then ∑ j = 0 n + 5 j + 5 2 - ∑ j = 0 n + 5 j + 5 is equal to ________.

Step-by-step solution

k = 1 2 9 - 1 2 + i 3 2 21 1 2 - 1 2 i 24 + 1 2 + i 3 2 21 1 2 + 1 2 i 24 k = 1 512 e i 2 π 3 21 e - i π 4 24 + e i π 3 21 e i π 4 24 ] k = 1 512 e i 14 π + 6 π + e i 7 π - 6 π k = 1 512 e 20 π i + e π i k = 1 512 [ 1 + - 1 ] = 0 n = k = 0 ∑ j = 0 5 j + 5 2 - ∑ j = 0 5 j + 5 = ∑ j = 0 5 j 2 + 25 + 10 j - j - 5 = ∑ j = 0 5 j 2 + 9 j + 20 = ∑ j = 0 5 j 2 + 9 ∑ j = 0 5 j + 20 ∑ j = 0 5 1 = 5 × 6 × 11 6 + 9 5 × 6