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

The pattern in the powers of i is determined by the fact that i is defined as the square root of -1. To find the value of large powers of i, we can use the pattern that emerges: i, -1, -i, 1. So to find the value of any large power of i, you can divide the exponent by 4 and use the remainder to determine which of the 4 values in the cycle it corresponds to.

Step-by-step explanation:

The pattern in the powers of i is determined by the fact that i is defined as the square root of -1. To find the value of large powers of i, we can use the pattern that emerges:

1. i1 = i
2. i2 = -1
3. i3 = -i
4. i4 = 1

As you can see, the powers of i repeat in a cycle of 4: i, -1, -i, 1. So to find the value of any large power of i, you can divide the exponent by 4 and use the remainder to determine which of the 4 values in the cycle it corresponds to. For example, i5 = i4 * i1 = 1 * i = i.

Answer 2

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`.