# How to evaluate expressions without explicitly computing a DFT?

I am trying to learn this part but I dont seem to understand. if I have a finite length sequence, and a n-point DFT of it (in an interval)... is it possible to evaluate expressions of the n-point DFT without computing them explicitly?

the exercise is:

Let $$x[n]$$ be finite-length sequence such that: $$x[n] = \{2,1,1,0,3,2,0,3,4,6\}$$ for $$0<=n<=9$$ and $$X[k]$$ be the 10-point DFT of $$x[n]$$ for $$0\le k\le9$$. evaluate the following without explicitly computing the DFT:

$$I) \space X$$ $$II) \space \sum_{k=0}^9 X[k]$$ $$III) \space \sum_{k=0}^9 |X[k]|^2$$ $$IV) \space \sum_{k=0}^9 X[k]e^{-j \frac {(4\pi)}{5}k}$$

and then it continues:

let $$X[k]$$ denote N-point DFT of an N-point sequence $$x[n]$$ ($$n$$ is even). Two N/2-point sequence are defined as:

$$g[n] = a_1 x[2n] + a_2 x[2n+1] \space where \space 0\le n\le(N/2)-1$$ $$h[n] = a_3 x[2n] + a_4 x[2n+1] \space where \space 0\le n\le(N/2)-1$$

constant $$a_i$$ satisfies: $$a_1 a_4 \neq a_2 a_3$$

let $$G[k]$$ and $$H[k]$$ denote the N/2-point DFT's of $$g[n]$$ and $$h[n]$$ respectively. Determine N-point DFT $$X[k]$$ in terms of $$G[k]$$ and $$H[k]$$ .

any clues on the solution?

any input is highly appreciated. thank you