Solution We need to show: in+1+in+2+in+3+in+4=0.Factor out in+1:in+1+in+2+in+3+in+4=in+1(1+i+i2+i3)Use i2=−1 and i3=−i:1+i+i2+i3=1+i+(−1)+(−i)=0Therefore:in+1(1+i+i2+i3)=in+1⋅0=0\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}We need to show: in+1+in+2+in+3+in+4=0.Factor out in+1:in+1+in+2+in+3+in+4=in+1(1+i+i2+i3)Use i2=−1 and i3=−i:1+i+i2+i3=1+i+(−1)+(−i)=0Therefore:in+1(1+i+i2+i3)=in+1⋅0=0