Q:

Solve for the upper limit of a summation

$a,b$ and $c$ are all natural numbers, and function $f(x)$ always returns a natural number. If$$
\sum_{n=b}^{a} f(n) = c,$$
in terms of $b,c$ and $f$, how would you solve for $a$? Do I require more information to solve for $a$?
EDIT: If $x$  increases $f(x)$ increases

A:

Once you add that $f(n)$ is increasing with $n$ you can set an upper bound by $\sum_{n=b}^{a} f(n) \ge (a-b+1)f(b)$.  As long as $f(b) \gt 0$ you can say $a \le \frac c{f(b)}+b-1$.  Now if you have a closed form you have brackets for your root finding, which can guarantee that you terminate.

