# Because roots of unity is too mainstream

**Algebra**Level 5

We further denote the summation \( \displaystyle P_m = \sum_{j=1}^n \left(a_j\right)^m \).

Find an expression for \(P_{nk} \).

Note: \(m,n,k\) are all positive integers.

**Bonus:** Can you write a result which provides all values of \(P_m\)?

**Clarification:** \(\Bbb C\) is the set of all complex values.

Inspired by this problem.