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 $\displaystyle z$ 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 $\displaystyle a + bi$.

$\displaystyle 1 - 2z + 4z^2 - 8z^3 = 0$

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

$\displaystyle \sum (-2z)^k$ from $\displaystyle k=0$ to $\displaystyle 3$

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

$\displaystyle \frac{a(1-r^n)}{1-r}$ = $\displaystyle \frac{-2(1-z^4)}{1-z}$

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 $\displaystyle z$, 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 $\displaystyle z^4 - 1 = 0$ or $\displaystyle z^4 = 1$. But when finding the four solutions to this equation (disregarding the solution $\displaystyle z=1$, as this would make the fraction indefinable), I do not get the right solutions.

To help, the actual solutions are $\displaystyle z = \frac{1}{2}, \pm \frac{1}{2}i$.

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

Thanks in advance.