Question

${\alpha }_1,{\alpha }_2,{\alpha }_3,........{\alpha }_{100}$ are all the 100th roots of unity. The numerical value of is
$\sum \sum _{1\le i<<j\le 100}({\alpha }_i{\alpha }_j{)}^5$

• A. $20$
• B. $0$
• C. $\left(20\right)\frac{1}{20}$
• D. none of these

Answer 1

The correct option is B $0$

$2\sum \sum _{1\le i<<j\le 100}({\alpha }_i{\alpha }_j{)}^5=({\alpha }_1^5+{\alpha }_2^5+{\alpha }_3^5+{\alpha }_4^5+{\alpha }_5^5+....{)}^2-({\alpha }_1^{10}+{\alpha }_2^{10}+{\alpha }_3^{10}+{\alpha }_4^{10}+{\alpha }_5^{10}+....)=0-0=0$

because $({\alpha }_1^r+{\alpha }_2^r+....+{\alpha }_{100}^r)=\{\begin{array}{ccc}100& if& r=100k\\ 0,& if& r\ne 100k\end{array}$