# Let $i=\sqrt{-1}.$ If  $\frac{\left ( -1+i\sqrt{3} \right )^{21}}{\left ( 1-i \right )^{24}}+\frac{\left ( 1+i\sqrt{3} \right )^{21}}{\left ( 1+i \right )^{24}}=k,$ and $n=\left [ \left | k \right | \right ]$ be the greatest integral part of $\left | k \right |.$ Then $\sum_{j=0}^{n+5}\left ( j+5 \right )^{2}-\sum_{j=0}^{n+5}\left ( j+5 \right )$ is equal to ________. Option: 1 310 Option: 2 310 Option: 3 340 Option: 4 340 Option: 5 350 Option: 6 350 Option: 7 360 Option: 8 360

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

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

$\\K=\frac{1}{512}\left[\mathrm{e}^{\mathrm{i}(14 \pi+6 \pi)}+\mathrm{e}^{\mathrm{i}(7 \pi-6 \pi)}\right] \\ \\\mathrm{K}=\frac{1}{512}\left[\mathrm{e}^{20 \pi \mathrm{i}}+\mathrm{e}^{\pi \mathrm{i}}\right]$

$\\K=\frac{1}{512}[1+(-1)]=0 \\ \\n=[|k|]=0$

$\\\sum_{\mathrm{j}=0}^{5}(\mathrm{j}+5)^{2}-\sum_{j=0}^{5}(\mathrm{j}+5) \\ \\\sum_{\mathrm{j}=0}^{5}\left(\mathrm{j}^{2}+25+10 \mathrm{j}-\mathrm{j}-5\right)$

$\\\sum_{j=0}^{5}\left(j^{2}+9 j+20\right) \\ \\\sum_{j=0}^{5} j^{2}+9 \sum_{j=0}^{5} j+20 \sum_{j=0}^{5} 1$

$\\\frac{5 \times 6 \times 11}{6}+9\left(\frac{5 \times 6}{2}\right)+20 \times 6 \\ \\=55+135+120 =310$

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