Roots of Unity

Roots of Unity: Level 3 Challenges


If w1w \not= 1 is an nn-th root of unity, then find the value of

1+w+w2+w3++wn1 1+w+w^2+w^3+\cdots +w^{n-1}


cosπ7cos2π7+cos3π7. \cos \frac{ \pi}{7} - \cos \frac{ 2\pi }{7} + \cos \frac{ 3 \pi}{7} .

x1729+x1729\huge x^{1729}+x^{-1729}

Find the value of the above expression if x+1x=1x + \frac 1 x = 1

If for zC ,z \in \mathbb{C} \space , z+1z=2cos6°.z+\dfrac{1}{z}=2 \cos 6°. Then find the value of (z1000+1z1000).\left( z^{1000}+\frac{1}{z^{1000}} \right).

Given z2+z+1=0,z^2+z+1=0, find the value of

(z+1z)2+(z2+1z2)2+(z3+1z3)2++(z21+1z21)2.\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.


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