11TH•Mathematics•Complex Numbers•Question 7

Solution

Show that $i^{n+1}+i^{n+2}+i^{n+3}+i^{n+4}=0$, for all $n\in\mathbb{N}$.

\begin{aligned} & \boxed{\text{We need to show: } i^{n+1}+i^{n+2}+i^{n+3}+i^{n+4}=0.} \\ \\ & \boxed{\text{Factor out } i^{n+1}:} \\ \\ & i^{n+1}+i^{n+2}+i^{n+3}+i^{n+4} \\ & = i^{n+1}(1+i+i^2+i^3) \\ \\ & \boxed{\text{Use } i^2=-1 \text{ and } i^3=-i:} \\ \\ & 1+i+i^2+i^3 = 1+i+(-1)+(-i)=0 \\ \\ & \boxed{\text{Therefore:}} \\ \\ & i^{n+1}(1+i+i^2+i^3)=i^{n+1}\cdot 0=0 \end{aligned}

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