Formula for $r+2r^2+3r^3+...+nr^n$

HINT: $r+2r^2+3r^3+... +nr^n=(r+r^2+\dots+r^n)+(r^2+r^3+\dots+r^n)+\dots+(r^n)$ and compute values in parentheses.

We have $$1+r+r^2 + \cdots + r^n = \dfrac{r^{n+1}-1}{r-1}$$ Differentiating this once, we obtain $$1+2r + 3r^2 + \cdots + nr^{n-1}= \dfrac{nr^{n+1}-(n+1)r^n + 1}{(r-1)^2}$$ Multiply the above by $r$ to obtain what you want.