Question

If \( z^2 + z + 1 = 0 \), where \( z \) is a complex number, then the value of \[ \left( z + \frac{1}{z} \right)^2 + \left( z^2 + \frac{1}{z^2} \right)^2 + \left( z^3 + \frac{1}{z^3} \right)^2 + \cdots + \left( z^6 + \frac{1}{z^6} \right)^2 \] is

• A. \( 18 \)
• B. \( 54 \)
• C. \( 6 \)
• D. \( 19 \)
• E. \( 12 \)

Hint:

If \( z^2 + z + 1 = 0 \), then \( z \) is a cube root of unity and powers repeat every 3 terms.

Answer 1

The Correct Option is C