I have the summation $$\sum_{k =1}^{n + 2}x^{n + 2 -k}y^k$$
which expands out to (taking $n = 3$ as an example)
$$x^4y^1 + x^3y^2 + x^2y^3 + x^1y^4.$$
How can I sum it? It looks similar to this question but the indices cannot be shifted to match that one.
It's a finite geometric sum with common ratio $y/x$. In general a finite geometric sum with common ratio $r \neq 1$ is "first term - first omitted term" divided by $1-r$, so in your case the result is $\frac{x^{n+1} y - x^{-1} y^{n+3}}{1-(y/x)}$ unless $y/x=1$.
