$\sum_{r=1}^{16}\left(\sin \frac{2 r \pi}{17}+i \cos \frac{2 r \pi}{17}\right)=$

One of the values of $(-64 i)^{5 / 6}$ is

If $\omega$ is a cube root of unity,then the value of $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5$ is:

If $n$ is an integer which leaves remainder $1$ when divided by $3$,then $(1+\sqrt{3}i)^n + (1-\sqrt{3}i)^n$ equals

If $\omega ( \neq 1)$ is a cube root of unity and $(1 + \omega )^7 = A + B\omega$,then $A$ and $B$ are respectively,the numbers

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 $n =[| k |]$ 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 ........ .