Could use help trying to find the following sum of series
$$ \sum_{n=1}^N r^n\sqrt{a + nd} $$
I have no clue where to begin on this one.
Ideally would like solution for all $ r $ but if it helps to assume $ r < 1 $ then we can start with assuming that.
There isn't going to be a 'nice' closed form, and as amcalde mentions, proving that seems hard. For the infinite series, we can write $$\sum_{n=1}^\infty r^n\sqrt{a+nd}=\sqrt{d}\Phi\left(r,-\frac{1}{2},\frac{a}{d}\right)-\sqrt{a},$$ where $\Phi$ is the Lerch Transcendent.