Given $z = cos \displaystyle \left ( \frac{2\pi}{2n + 1} \right ) + i sin \left ( \frac{2\pi}{2n + 1} \right )$, n a positive integer, find the equation whose roots are $\alpha = z+ z^3 + ...... + z^{2n - 1}$ and $\beta = z^2 + z^4 + ..... + z^{2n}$
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$z^2 -z + \displaystyle \frac{1}{4} sec^2 \left ( \frac{\pi}{2n + 1} \right )$
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$z^2 +z + \displaystyle \frac{1}{4} sec^2 \left ( \frac{\pi}{2n + 1} \right )$
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$z^2 +z + \displaystyle \frac{1}{4} sec^2 \left ( \frac{\pi}{2n } \right )$
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$z^2 +z + \displaystyle \frac{1}{2} sec^2 \left ( \frac{\pi}{2n +1} \right )$
B
Correct answer
Explanation
This is a complex roots of unity problem. Given the structure of alpha and beta as sums of powers of z, they are roots of a quadratic equation derived from the properties of trigonometric sums. The correct form is z^2 + z + 1/4 * sec^2(pi / (2n+1)).