Solve this following

Question:

Let $\mathrm{i}=\sqrt{-1}$. If $\frac{(-1+\mathrm{i} \sqrt{3})^{21}}{(1-\mathrm{i})^{24}}+\frac{(1+\mathrm{i} \sqrt{3})^{21}}{(1+\mathrm{i})^{24}}=\mathrm{k}$,

and $\mathrm{n}=[|\mathrm{k}|]$ be the greatest integral part of

$|\mathrm{k}|$. Then $\sum_{\mathrm{j}=0}^{\mathrm{n}+5}(\mathrm{j}+5)^{2}-\sum_{\mathrm{j}=0}^{\mathrm{n}+5}(\mathrm{j}+5)$ is equal to

 

 

Solution:

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

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

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

$\mathrm{K}=\frac{1}{512}\left[\mathrm{e}^{20 \pi \mathrm{i}}+\mathrm{e}^{\pi \mathrm{i}}\right]$

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

$\mathrm{n}=[|\mathrm{k}|]=0$

$\sum_{j=0}^{5}(j+5)^{2}-\sum_{j=0}^{5}(j+5)$

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

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

$\sum_{j=0}^{5} j^{2}+9 \sum_{j=0}^{5} j+20 \sum_{j=0}^{5} 1$

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

$=55+135+120$

$=310$

 

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