If z2+z+1=0, where z is a complex number, then the value of (z+1z)2+(z2+1z2)2+(z3+1z3)2+....+(z6+1z6)2 is

z2+z+1=0⇒z=ω or ω2 G.E=(ω+1ω)2+(ω2+1ω2)2+(ω3+1ω3)3+....(ω6+1ω6)6 =4(ω+ω2)2+2(ω3+ω3)2=12.

If z2+z+1=0, where z is a complex number, then the value of (z+1z)2+(z2+1z2)2+(z3+1z3)2+....+(z6+1z6)2 is

z2+z+1=0⇒z=ω or ω2 G.E=(ω+1ω)2+(ω2+1ω2)2+(ω3+1ω3)3+....(ω6+1ω6)6 =4(ω+ω2)2+2(ω3+ω3)2=12.

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If $z^{2} + z + 1 = 0,$ where $z$ is a complex number, then the value of
${(z + \frac{1}{z})}^{2} + {(z^{2} + \frac{1}{z^{2}})}^{2} + {(z^{3} + \frac{1}{z^{3}})}^{2} + .... + {(z^{6} + \frac{1}{z^{6}})}^{2}$ is

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Detailed Solution

$z^{2} + z + 1 = 0 \Rightarrow z = ω o r ω^{2} G . E = {(ω + \frac{1}{ω})}^{2} + {(ω^{2} + \frac{1}{ω^{2}})}^{2} + {(ω^{3} + \frac{1}{ω^{3}})}^{3} + .... {(ω^{6} + \frac{1}{ω^{6}})}^{6} = {4 (ω + ω^{2})}^{2} + {2 (ω^{3} + ω^{3})}^{2} = 12.$