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My Thoughts are: Using geometric series Sn= (a×(r^(n+1) -1) ÷ (r-1) ) for a=1 , r=2i , n=26 Now as i know; i^2=-1 , so i^27 = i^3 =-1 => (2i)^27 = (2^27) × i^27 = -2^27 ×i => (1-2^27 × i) ÷ (-3) = -1/3 - 2i/3 So Re(z) = -1/3 , Im(z)= -2^27/3 But my problem then how l'm i gonna find |z| ? According to the def. |z|= √(-1/3)+(-2^(27)/3)^2 Where did i go wrong? Should i use a diffrent formulare ?

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My Thoughts are: Using geometric series $S_n= a(r^{n+1} -1) /(r-1) )$ for $a=1 , r=2i , n=26$ Now as I know that $i^2=-1$ , so $i^{27} = i^3 =-1$ $\implies (2i)^{27} = (2^27) × i^{27} = -2^{27} i$ $\implies (1-2^{27} i)/(-3) = -1/3 - 2i/3 $
So $Re(z) = -1/3$ , $Im(z)= -2^{27}/3$. But my problem then how l'm i gonna find $|z|$ ? According to the def. $|z|= √(-1/3)+(-2^{27}/3)^2$ Where did I go wrong? Should i use a different formula?