For an arbitrary, finite sequence of real numbers, does an explicit formula always exist? For a finite sequence of real numbers An={a1, a2, a3,..., an}, will there always be an explicit formula to describe it?
So for example, suppose I choose "at random" the sequence of numbers: -3, 1904,  17, 1/12, 21245, -3/16. Is it guaranteed that an explicit formula exists that will generate each of the terms - even though it may be a huge and incredibly complex? 

Yes. For a sequence of $n$ values, $a_1, a_2, \ldots, a_n$ there's not just a formula...there's actually a polynomial, $p$, and it has degree at most $n-1$, with the property that 
$$
p(1) = a_1\\
p(2) = a_2 \\
\ldots\\
p(n) = a_n
$$
For instance, for a sequence of two values, there's a linear polynomial, whose graph is the straight line joining the two points $(1, a_1)$ and $(2, a_2)$.
This polynomial is called the Lagrange Interpolating polynomial or Lagrange interpolant. See this reference. The formula for the polynomial, in terms of the $a_i$ values, is a little messy, but not horrible. 
Most articles on the lagrange interpolant want a sequence of pairs $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$. In your case, each $x_i$ is just $i$, i.e. your pairs are $(1, a_1), (2, a_2), \ldots, (n, a_n)$. 
