Complex numbers : finding real & imaginary numbers and the magnitude

Question

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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?

Answer 1

First of all $i^{27}=-i$ but I think that you just had a finger error. Then you have that $z=\frac{r^{n+1}-1}{r-1}$ with $r=2i$, then \begin{align*} z=\frac{-2^{27}i-1}{2i-1}&= \frac{-2^{27}i-1}{2i-1}\cdot \frac{2i+1}{2i+1}\\ &=\frac{2^{28}-1-(2^{27}+2)i}{-5} \end{align*} So, $Re(z)=\frac{2^{28}-1}{-5}$, $Im(z)=\frac{(2^{27}+2)i}{5}$ and \begin{align*} |z|&=\sqrt{\left(\frac{2^{28}-1}{-3}\right)^2+\left(\frac{(2^{27}+2)i}{5}\right)^2}\\ &=\sqrt{\frac{2^{56}+2^{54}+5}{25}} \end{align*}