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 \( k \).
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 \( k \).
Show Hint
Use the properties of roots of unity to simplify the powers and compute the terms effectively.
Updated On: May 15, 2025
- \( r, r \in \mathbb{N} \)
- \( 2r+1, r \in \mathbb{N} \)
- \( 4r+1, r \in \mathbb{N} \)
- \( 3r, r \in \mathbb{N} \)
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The Correct Option is A
Solution and Explanation
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.
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