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

7-110.

Consider this geometric sequence:

*i*^{0},*i*^{1},*i*^{2},*i*^{3},*i*^{4},*i*^{5}, …,*i*^{15}. Homework Help ✎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.Use the pattern you found in part (a) to calculate

*i*^{16},*i*^{25},*i*^{39}, and*i*^{100}.What is

*i*^{4}, where^{n}*n*is a positive integer?Based on your answer to part (c), simplify

*i*^{4}^{n}^{+1},*i*^{4}^{n}^{+2}, and*i*^{4}^{n}^{+3}.Calculate

*i*^{396},*i*^{397},*i*^{398}, and*i*^{399}.

*i*^{3} = *i*^{2}·*i*

*i*^{4} = *i*^{2}· *i*^{2}

*i*^{25} = *i*^{4(6) + 1} = (*i*^{4})^{6}*i*^{1} = ?

*i* ^{4}* ^{n}* = (

*i*

^{4})

*= 1*

^{n}

^{n}*i*, −1, −*i*

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