Question

${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,.........{\alpha }_{100},$ are all the ${100}^{th}$ roots of unity. Then the numerical value of ${\sum }_{1\le i}{\sum }_{j\le 100}({\alpha }_i{\alpha }_j{)}^5$ is

• A. 20
• B. 0
• C. $(20{)}^{1/20}$
• D. None of these

Answer 1

The correct option is B 0

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