Question about Geometric Series and Roots of Unity

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.

Re: Question about Geometric Series and Roots of Unity

why should we complicate the situation by going in for GP. Just group the terms and we get

( 1 - 2z ) + 4z^2 ( 1 - 2z ) = ( 1-2z)( 1 + 4z^2)=0

that gives z = 1/2 and z^2= -1/4 thus z = + 0r - i/2

Now for the GP

The formula is right

Sum = { 1 - ( -2z)^4}/ { 1 - ( -2z)} = ( 1 - 16z^4)/(1+2z) = { ( 1+4z^2)(1+2z)(1-2z) / ( 1+ 2z) } = ( 1+4z^2)(1-2z) That is the same what we got earlier.

Re: Question about Geometric Series and Roots of Unity

Thank you very much. Yeah, when I did the sum, I didn't realise 'r' was '(-2z)' and not just 'z' with -2 factored out. That makes sense. :)

Re: Question about Geometric Series and Roots of Unity

If you are going to sum the geometric series, get your a and r right. Your and your , so the sum is actually .

So solving your equation:

But since as it would give a 0 denominator, we must disregard this answer.