complex polynomial

Dec 2015
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128
Earth
Compute Z in terms of N .
\(\displaystyle z=i+i^2 +i^3 +\dotsc + i^N \; , \: N\in \mathbb{N}.\)
 

romsek

Math Team
Sep 2015
2,885
1,609
USA
there's perhaps a fancier formula but

$z_N = \begin{cases}i &N \pmod {4}=1\\-1 + i &N \pmod{4}=2\\-1 &N \pmod{4} = 3\\0 &N\pmod{4}=0\end{cases}$
 

skipjack

Forum Staff
Dec 2006
21,387
2,410
\(\displaystyle \frac{i^N - 1}{i + 1}\)
 
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