Let z =(1-i √3/2), i=√-1. Then the value of 21+(z+(1/z))3+(z2+(1/z2))3+(z3+(1/z3))3+ ldots+(z21+(1/z21))3 is

Question

Let z =(1-i √3/2), i=√-1. Then the value of 21+(z+(1/z))3+(z2+(1/z2))3+(z3+(1/z3))3+ ldots+(z21+(1/z21))3 is  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Z =(1-√3 i /2)= e - i (π/3) z r +(1/ z r )=2 cos (-(π/3)) r =2 cos ( r π/3) ⇒ 21+ displaystyle∑ r =121( z r +(1/ z r ))3=8( cos 3 ( r π/3))=2( cos r π+3 cos ( r π/3)) ⇒ 21+( z +(1/2))3+( z 2+(1/ z 2))3+ ldots ldots( z 21+(1/ z 21))3 =21+ displaystyle∑ r =121( z r +(1/ z r ))3 =21+ displaystyle∑ r =121(2 cos r π+6 cos ( r π/3)) =21-2-6 =13

Q. Let z=21−i3​​,i=−1​. Then the value of 21+(z+z1​)3+(z2+z21​)3+(z3+z31​)3+…+(z21+z211​)3 is _____

Z=21−3​i​=e−i3π​ zr+zr1​=2cos(−3π​)r=2cos3rπ​ ⇒21+r=1∑21​(zr+zr1​)3=8(cos33rπ​)=2(cosrπ+3cos3rπ​) ⇒21+(z+21​)3+(z2+z21​)3+……(z21+z211​)3 =21+r=1∑21​(zr+zr1​)3 =21+r=1∑21​(2cosrπ+6cos3rπ​) =21−2−6 =13

Q. Let $z = \frac{1 - i 3}{2}, i = - 1$ . Then the value of $21 + (z + \frac{1}{z})^{3} + (z^{2} + \frac{1}{z ^{2}})^{3} + (z^{3} + \frac{1}{z ^{3}})^{3} + \dots + (z^{21} + \frac{1}{z ^{21}})^{3}$ is _____