discrete signals - DFT: a function of $n$?

I‘m a high school student and I haven’t studied physics or anything.

Why does the DFT depend on an integer, say $k$ or $n$ (it’s usually expressed like $F(n)=...$ or $F(k)$ or $F_k$, etc.) if it is supposed to deliver a frequency information of a sampled signal?

Can the frequency content of the signal be expressed as a multiple of the integer?

Answer

Let us assume that you have a finite length discrete signal $x$, denoted by its samples $x_n$, $0\le n; $x$ does not depend on $n$, but its is values are indexed by $n$. Once you index a signal with integers, it somehow "looses" its dependence to an "actual time" in seconds. In other words, one does not know how much time actually elapsed between $x_{13}$ and $x_{14}$. And, in a relative way, one does not care, when it comes to understanding which (relative) frequencies compose $x$.

When we compute the DFT of $x$, we turn its $N$ values onto $K$ other values $F_k$ (most often $K=N$), indexed by $0\le k. The $F_k$'s are Fourier amplitudes, relatively indexed by integers, but the Fourier transform, globally, does not depend on an integer.