Question

Examine the pattern in the powers of i you wrote in the table, and create a rule for finding the value of large powers of i. Justify your answer.

Answer 1

Let $N$ be any integer. Then

$i^N=\begin{cases}1&\text{if }N\equiv0\mod4\\i&\text{if }N\equiv1\mod4\\-1&\text{if }N\equiv2\mod4\\-i&\text{if }N\equiv3\mod4\end{cases}$

In other words:

(1) If $N$ is a multiple of 4, then $i^N=1$.

(2) If dividing $N$ by 4 leaves a remainder of 1, then $i^N=i$, since $i^N=i^{4n+1}$ for some integer $n$, and from (1) we know that $i^{4n+1}=i^{4n}i=i$ (because, obviously, $4n$ is a multiple of 4).

(3) If instead you get a remainder of 2, then $i^N=-1$. This follows from (1) as well. $i^N=i^{4n+2}=i^{4n}i^2=(1)(-1)=-1$.

(4) Finally, if you get a remainder of 3, then $i^N=i^{4n+3}=i^{4n}i^2i=(1)(-1)(i)=-i$.