Question

a₁,a_­2, a₃,......., a₁₀₀ are all 100^th roots of unity. The numerical value of $\sum_{1 \leq i}^{}{\sum_{< j \leq 100}^{}{(\alpha_{i}\alpha_{j})^{5}}}$ is-

• A. 20
• B. 0
• C. (20)^1/20
• D. 100

Answer 1

Answer: (B)

0

Answer 2

Sol. $\sum_{1 \leq i}^{}{\sum_{< j \leq 100}^{}{\alpha_{i}^{5}\alpha_{j}^{5}}}$= ($\alpha_{1}^{5}$ + $\alpha_{2}^{5}$+ $\alpha_{3}^{5}$+.......)² – ($\alpha_{1}^{10}$+$\alpha_{2}^{10}$+......)

By using = 0 – 0 = 0

$\alpha_{1}^{r}$+ $\alpha_{2}^{r}$+.......+$\alpha_{100}^{r}$= 100 if r = 100k

= 0 if r ¹ 100 k