You can simplify the equation by using the fact that :
i^2 = -1
i^3 = (i)(i^2)=-i
i^4 = (i^2)^2 = 1
i^5 = (i^4) (i)=i
and so on
songoku's method is the fastest. You need to recognize the fact that the powers of i cycle through four different numbers: 1, i, -1, and -i. The sum of any four consecutive powers of i will be 0. The fraction could be written like this:
The sum of the first four powers of i shown above is
So how many powers of i are being added all together? What is the sum in the numerator? If you can answer these questions, you'll know what the fraction simplifies to.
01