Question

Let $i=\sqrt{-1}.$ If $\frac{{(-1+i\sqrt{3})}^{21}}{(1-i{)}^{24}}+\frac{{(1+i\sqrt{3})}^{21}}{(1+i{)}^{24}}=k,$ and $n=\left[|k|\right]$ be the greatest integral part of $|k|.$ Then $\sum _{j=0}^{n+5}(j+5{)}^2-\sum _{j=0}^{n+5}(j+5)$ is equal to

Answer 1

$\frac{{(-1+i\sqrt{3})}^{21}}{(1-i{)}^{24}}+\frac{{(1+i\sqrt{3})}^{21}}{(1+i{)}^{24}}$
$=\frac{{\left(2e^{i\frac{2\pi }{3}}\right)}^{21}}{{\left(\sqrt{2}{e}^{-i\frac{\pi }{4}}\right)}^{24}}+\frac{{\left(2e^{i\frac{\pi }{3}}\right)}^{21}}{{\left(\sqrt{2}{e}^{i\frac{\pi }{4}}\right)}^{24}}$
$=\frac{2^{21}\left(e^{i14\pi }\right)}{2^{12}\left(e^{-i6\pi }\right)}+\frac{2^{21}\left(e^{i7\pi }\right)}{2^{12}\left(e^{i6\pi }\right)}$
$=2^9{e}^{i\left(20\pi \right)}+2^9{e}^{i\pi }$
$=2^9+2^9(-1)=0=k$
$\therefore n=0$

$\sum _{j=0}^5(j+5{)}^2-\sum _{j=0}^5(j+5)$
$=[5^2+6^2+7^2+8^2+9^2+{10}^2]-[5+6+7+8+9+10]$
$=\left[\right(1^2+2^2+\cdots +{10}^2)-(1^2+2^2+3^2+4^2\left)\right]-\left[\right(1+2+\cdots +10)-(1+2+3+4\left)\right]$
$=(385-30)-(55-10)$
$=355-45=310$