Question

If \( \omega, \omega^2 \) are the cube roots of unity, \( k \) is a positive integer, and \[ (1 - \omega^2)^{3k} + (1 - \omega^2 + 0)^{3k+1} + (1 + \omega - 0)^{3k+1}, \] then find \( k \).

• A. \( r, r \in \mathbb{N} \)
• B. \( 2r+1, r \in \mathbb{N} \)
• C. \( 4r+1, r \in \mathbb{N} \)
• D. \( 3r, r \in \mathbb{N} \)

Hint:

Use the properties of roots of unity to simplify the powers and compute the terms effectively.

Answer 1

The Correct Option is A

We are given a recurrence involving cube roots of unity and need to find the value of \( k \) based on the conditions. The roots of unity satisfy certain properties that we use to simplify the given expression and find the value of \( k \). This involves solving the relation step by step.