Consider this geometric sequence: $i^0, i^1, i^2, i^3, i^4, i^5 ,...,i^{15}$. .

1. You know that $i^0 = 1$, $i^1 = i$, and $i^2 = -1$. Calculate the result for each term up to $i^{15}$, and describe the pattern.

   Hint (a):

   $i^0 = 1$ Can you see the pattern?
   $i^1 = i$
   $i^2 = -1$
   $i^3 = (i)(i^2) = -i$
   $i^4 = (i^2)(i^2) = 1$
   $i^5 = (i^2)(i^2)(i) = i$

   Answer (a):

   The pattern repeats $1$, $i$, $-1$, and $-i$.

2. Use the pattern you found in part (a) to calculate $i^{16}, i^{25}, i^{39},$ and $i^{100}$.

   Hint (b):

   $16$ is a multiple of $4$.
   $25$ is one more than a multiple of $4$.
   $39$ is one less than a multiple of $4$.
   $100$ is a multiple of $4$.

   Answer (b):

   $i^{16} = 1$
   $i^{25} = i$
   $i^{39} = -i$
   $i^{100} = 1$

3. What is $i^{4n}$ , where $n$ is a positive whole number?

   Hint (c):

   $4n$ is a multiple of $4$.

4. Based on your answer to part (c), simplify $i^{4n+1}$, $i^{4n+2}$, and $i^{4n+3}$.

   Hint (d):

   Refer to (a), (b), and (c).

5. Calculate $i^{396}$, $i^{397}$, $i^{398}$, and $i^{399}$.

   Hint (e):

   $i^{396} = i^{4 · 99} = 1$
   $i^{397} = i^{4 · 99+1}=?$
   $i^{398} = i^{4 · 99 + 2} = ?$
   $i^{399} = ?$