What is the DFT of $[x_1 -x_2 x_3 -x_4....x_n]$
Hint: Consider a factor 2 down-sampled sequence : $\tilde{x} = \{x_1, x_3, x_5, ...\}$ and consider another sequence which is circular shifted by $-1$ sample and downsampled by a factor of 2 : $\hat{x} = \{ x_2, x_4, x_6, ...\}$. Observe that the required sequence is difference of $\tilde{x}$ and $\hat{x}$. Each of the DFT can be computed by observation using properties.