  For a positive integer $d$, let $p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d$;
i.e., $p_d(n)$ is the sum of the first $d^{\mathrm{th}}$-powers up to $n$. It's
well known that $p_d(n)$ is a polynomial of degree $d+1$ in $n$. While this is
usually proved by induction, once $d$ is not small it's a challenge as one
needs to know the polynomial for the inductive step. We show how this
difficulty can be bypassed by giving a simple proof that $p_d(n)$ is a
polynomial of degree $d+1$ in $n$ by using L'Hopital's rule, and show how we
can then determine the coefficients by Cramer's rule. This illustrates a
general principle and the point of our paper: there's more than one path to a
goal, different approaches have their advantages and disadvantages, and the
more techniques one knows, the more likely one can successfully attack a
problem.
