Let i = - 1 . If (- 1 + i 3 ) ^ 21 (1 - i) ^ 24 + (1 + i 3 ) ^ 21...

$k = \dfrac{1}{{2}^{9}}\left[\dfrac{{\left(- \dfrac{1}{2} + \dfrac{i\sqrt{3}}{2}\right)}^{21}}{{\left(\dfrac{1}{\sqrt{2}} - \dfrac{1}{\sqrt{2}}i\right)}^{24}} + \dfrac{{\left(\dfrac{1}{2} + \dfrac{i\sqrt{3}}{2}\right)}^{21}}{{\left(\dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}i\right)}^{24}}\right]$

$k = \dfrac{1}{512}\left[\dfrac{{\left({e}^{i\dfrac{2\pi}{3}}\right)}^{21}}{{\left({e}^{- \dfrac{i\pi}{4}}\right)}^{24}} + \dfrac{{\left(e\dfrac{i\pi}{3}\right)}^{21}}{{\left(e\dfrac{i\pi}{4}\right)}^{24}}\right ]$

$k = \dfrac{1}{512}\left[{e}^{i\left(14\pi + 6\pi\right)} + {e}^{i\left(7\pi - 6\pi\right)}\right]$

$k = \dfrac{1}{512}\left[{e}^{20\pii} + {e}^{\pii}\right]$

$k = \dfrac{1}{512} [ 1 + \left(- 1\right) ] = 0$

$n = \left[\left|k\right|\right] = 0$

$\mathop{∑}_{j = 0}^{5}{\left(j + 5\right)}^{2} - \mathop{∑}_{j = 0}^{5}\left(j + 5\right)$

$= \mathop{∑}_{j = 0}^{5}\left({j}^{2} + 25 + 10j - j - 5\right)$

$= \mathop{∑}_{j = 0}^{5}\left({j}^{2} + 9j + 20\right)$

$= \mathop{∑}_{j = 0}^{5}{j}^{2} + 9\mathop{∑}_{j = 0}^{5}j + 20\mathop{∑}_{j = 0}^{5}1$

$= \dfrac{5 \times 6 \times 11}{6} + 9\left(\dfrac{5 \times 6}{2}\right) + 20 \times 6$

$= 55 + 135 + 120$

$= 310$