### Home > INT3 > Chapter 7 > Lesson 7.2.5 > Problem7-110

7-110.

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.

$i^{3} = i^{2}·i$
$i^{4} = i^{2}· i^{2}$

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

$i^{25} = i^{4\left(6\right) + 1} = \left(i^{4}\right)^{6}i^{1} = ?$

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

$i^{4n} = (i^4)^n = 1^n$

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

$i, −1, −i$

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

Use the same pattern you found in part (a).