Solution Let n∈N, n>4. Write n=4q+r, 0≤r<4.Then:in=i4q+r=i4q ir=(i4)q irSince i4=1:(i4)q=1q=1⇒in=ir\begin{aligned} & \boxed{\text{Let } n\in\mathbb{N},\ n>4.\ \text{Write } n=4q+r,\ 0\le r<4.} \\ \\ & \boxed{\text{Then:}} \\ \\ & i^n = i^{4q+r} = i^{4q}\,i^r = (i^4)^q\,i^r \\ \\ & \boxed{\text{Since } i^4=1:} \\ \\ & (i^4)^q = 1^q = 1 \\ \\ & \Rightarrow i^n = i^r \end{aligned}Let n∈N, n>4. Write n=4q+r, 0≤r<4.Then:in=i4q+r=i4qir=(i4)qirSince i4=1:(i4)q=1q=1⇒in=ir Also: r=0⇒in=1r=1⇒in=ir=2⇒in=−1r=3⇒in=−i\begin{aligned} & \boxed{r=0\Rightarrow i^n=1} \\ & \boxed{r=1\Rightarrow i^n=i} \\ & \boxed{r=2\Rightarrow i^n=-1} \\ & \boxed{r=3\Rightarrow i^n=-i} \end{aligned}r=0⇒in=1r=1⇒in=ir=2⇒in=−1r=3⇒in=−i