Finding alternating sum from DFT of length-7 vector

[bodensee9]

Homework Statement

Hello: I have a say N = 7 vector. Say I know that it's DFT is X = [X[0] X[1] ... X[6]]
I am wondering if there is any way that I can find the sum of:
x[1] - x[2] + x[3] - x[4] + x[5] - x[6]?
Since N = odd, I can't use the trick with N/2 = pi as a frequency. Thanks so much.
My signal x{n] is real, but I really can't see any relation between the entries of X and x.

 

[marcusl]

You could express your modified sequence as the product of the sequence x times a square wave:

$$x'_n=(-1)^n x_n,$$ with n=0, 1, ...

Try taking it from there. Hint: what is the Fourier transform of a product of two functions?