- #1

Olinguito

- 239

- 0

$$A\circ B\ =\ \{z_1z_2:z_1\in A,\,z_2\in B\}.$$

Further define $A^{[1]}=A$ and recursively $A^{[n]}=A^{[n-1]}\circ A$ for $n>1$.

Let $\zeta_n=\{z\in\mathbb C:z^n=1\}$. Given a fixed integer $n\geqslant2$ and any positive integer $r$, find the sum of all the elements in $\zeta_n^{[r]}$.