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im stuck... and this is the last question... Oct 7, 2019

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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   Oct 7, 2019