+128474 Here's all you reaaly need to know here
If we divide the power on i by 4 and we get a remainder of 1, the result is i
If we divide the power on i by 4 and get a remainder of 2, the result is -1
If we divide the power on i by 4 and get a remainder of 3, the result is - i
If we divide the power on i by 4 and get a whole number, the result is 1
So
i^4 = 4/4 = 1 = a whole number....so....the result is 1
i^5 = 5/4 = 1 + 1/4 = remainder of 1, so the result is i
i^6 = 6/4 = 1 + 2/4 = remainder of 2, so the result is -1
i^7 = 7/4 = 1 + 3/4 = remainder of 3, so the result = -i
i^8 = 8/4 = 2....whole number ....result is 1
i^9 = 9/4 = 2 + 1/4 = remainder = 1 so the result = i
i^10 = 10/4 = 2 + 2/4 = remainder of 2 ....so the result is -1
b. The pattern is, for any integer n ≥ 1
i^(4n-3) = i
i^(4n - 2) = -1
i^(4n - 1) = -i
i^(4n) = 1
c. i^2001 = 2001/4 = 500 + 1/4 ....remainder = 1....so....the result = i
