Hello. I am working on my own through the AQA FP2 Mathematics textbook (A2-Level) and I have come across a question I am stuck on. I am given the following equation, where is a complex number, and asked to solve the equation, in other words find the roots of the equation. My answers should be in the form .

Initially I noticed that this is a geometric series and can therefore be expressed as a summation:

But using the related formula for a geometric series I get:

=

note: n=4 because the final term of a sum of a geometric series is always n-1 (3).

However, when I substitute a random number in for , say 2, the two expressions do not equate. So somewhere, my geometric series expression is incorrect, but I cannot see where. I have not done geometric series in a while.

Continuing anyway, I would then go on to say that the roots of the initial equation must be three of the roots of or . But when finding the four solutions to this equation (disregarding the solution , as this would make the fraction indefinable), I do not get the right solutions.

To help, the actual solutions are .

If anyone could help me see what I have done wrong with the geometric series, I would be very grateful.

Thanks in advance.