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what is the sum of i^2017 , i ^2018 , and i ^2019 where i = square root of -1?

 Nov 5, 2020
 #1
avatar+111600 
+2

Just look for the pattern

 

\(i^1=i\\ i^2=-1\\ i^3=-1i\\ i^4=+1\\ i^5=i\\ \text{and so the pattern starts over.} \)

 

Use the pattern to determine your answers.

 Nov 5, 2020
 #2
avatar+25597 
+3

what is the sum of \(i^{2017}\) , \(i ^{2018}\) , and \(i ^{2019}\) where i = square root of -1?

 

\(Formula: \boxed{\mathbf{i^n+i^{2+n}=0}}\)

 

 

\(\begin{array}{|rcll|} \hline && i^{2017}+i^{2018}+i^{2019} \\ &=& i^{2017}+i^{2019}+i^{2018} \quad | \quad \mathbf{i^{2017}+i^{2019} = 0} \\ &=& 0 + i^{2018} \\ &=& i^{2*1009} \\ &=& \left(i^{2}\right)^{1009} \quad | \quad \mathbf{i^2 = -1} \\ &=& \left(-1\right)^{1009} \\ &=& \mathbf{-1} \\ \hline \end{array}\)

 

laugh

 Nov 5, 2020

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