Frequency of a DFT term

I've been wrestling with this question for way too long. I appreciate any insight you can offer. Thanks.

If the $9$-point sample sequence $x(n)$ represents a time domain signal sampled at $48\textrm{ kHz}$, what frequency does the eighth term $X[7]$ in its DFT correspond to?

If you have a signal of 9 samples, namely ${\left\{ {x}_{n} \right\}}_{n = 0}^{8}$ its DFT is given by ${\left\{ {X}_{k} \right\}}_{k = 0}^{8}$

Now if the sampling rate is ${F}_{s}$ the bin frequency interval is given by $\frac{{F}_{s}}{N}$.

Now, you have 9 samples, 1 is the DC ($k = 0$) and the rest spans $\left[ \frac{-{F}_{s}}{2}, \frac{{F}_{s}}{2} \right]$.

You can look on the first 5 which are $\left\{ 0, \frac{{F}_{s}}{9}, \frac{2 {F}_{s}}{9}, \frac{3 {F}_{s}}{9}, \frac{4 {F}_{s}}{9} \right\}$.

Now, the 7 bin would have to be symmetric and hence would be $\frac{-3 {F}_{s}}{9}$.

• Dan, please mark the question as answered. Thank You. – Royi Aug 9 '15 at 16:43