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|.

Question

Let i = √-1. If \(\frac{(-1 + i\sqrt3)^{21}}{(1-i)^{24}}\) + \(\frac{(1 + i\sqrt3)^{21}}{(1+i)^{24}}\) = k, and n = [|k|] be the greatest integral part of |k|. Then \(\displaystyle\sum_{j=0}^{n+5} (j + 5)^2\) - \(\displaystyle\sum_{j=0}^{n+5} (j + 5)\) is equal to

Answer 1

\(\frac{5 \times 6 \times 11}{6} + 9(\frac{5 \times 6}{2}) + 20 \times 6\)

= 55 + 135 + 120

= 310