#1**+1 **

a) i^4 = 1, i^5 = i, i^6 = -1, i^7 = -i, i^8 = 1, i^9 = i, i^10 = -1.

b) The explanation is the the pattern cycles in powers of 4. Every 4th power (like i^4 or i^8) is always 1. Anything 1 above a multiple of four (ex. i^9) is always just some multiple of four multiplied by i, so it would be i. For anything two above, it would be a multiple of 4 * i^2, so -1. For i^3*multiple of 4, it would be -1. And then i^4 * multiple of 4 is 1 and we cycle again.

c) We know that i^2000 = 1, and therefore i^2001 = i^2000*i = 1*i = i.

Hope this helped!

OlympusHero Dec 23, 2019