I'm not sure if this equation is a linear homogeneous recurrence relation because I didn't learn math in english. what is the explicit formula for $f$?
$$f(x) = f(x-2)-{n^{x-1}\over NW-1} f(x-1)+n^{x-1}$$
$n$ and $NW$ are constants. $n = 3$ and $NW = 7000000000 = $7 billion I'm having a hard time because of the $n^{x-1}$ please help me.
This is a linear second order recurrence with variable coefficients. Just as in the case of differential equations, there are few such recurrences with closed form solutions.
As a first approximation (for smallish $x$) you can disregard the term in $f(x - 1)$, and get a first order linear recurrence (of step two) in $f(x)$:
$$ f(x) - f(x - 2) = n^{x - 1} $$
This has solution:
$$ f(2 x) = f(0) + \sum_{0 \le k \le x} n^{k - 1} = f(0) + \frac{n^{x + 1} - 1}{n (n - 1)} $$
This might be enough for your purposes, or use this as a starting point to construct a closer approximation.