$i+i^2+i^3+i^4+......+i^{100}$

$$\begin{gathered} \begin{array}{r}{i}^n+i^{n+1}+i^{n+2}+i^{n+3}=i^n\left[1+i+i^2+i^3\right]\\ =i^n[1+i-1-i]\\ =i^n\left(0\right)=0\end{array} \\ \Rightarrow \boxed{i^n+i^{n+1}+i^{n+2}+i^{n+3}=0\forall n\in N} \end{gathered}$$

$$\begin{gathered} i+i^2+i^3+i^4+......+i^{100} \\ =\left(i+i^2+i^3+i^4\right)+\left(i^5+i^6+i^7+i^8\right)+......+\left(i^{97}+i^{98}+i^{99}+i^{100}\right) \\ =0 \end{gathered}$$