Fourier transform of particles on a loaded string

[Mrbluesky323]

Homework Statement

Can someone tell me how to Fourier transform this quantity:
$$\Sigma$$ (x_(j+1) - x_j)^2

where the sum is from j=1 to N

Homework Equations

Define the Fourier transform as
x_j = $$\Sigma$$ A_k *exp(-iqkj)
**Where i is sqrt(-1)
**The Sum is from k=0 to (N-1)
**q = (2*pi)/N
**Assume periodic boundary conditions (i.e. x_(N+1) = x_1)

The Attempt at a Solution

I get to

$$\Sigma$$_j ($$\Sigma$$_k A_k* exp(-iqkj) (exp(-iqk) - 1))^2

I'm told the answer one arrives at ought to be:

$$\Sigma$$_k A_k* A_-k*(2sin(qk/2))^2
I'm not sure how to get there from where i am.
Any help would be appreciated.

 

[Coto]

As a first step, factor out $e^{-iqk/2}$ from the expression $e^{-iqk}-1$ and note the identity:
$$e^{iqk/2} - e^{-iqk/2} = 2\sin{qk/2}$$