jmc

Let $S=r^2+2r^5+3r^8+4r^{11}+\cdots .$ Then $$r^{3}S=r^5+2r^8+3r^{11}+4r^{13}+\cdots .$$Subtracting this equation from $S=r^2+2r^5+3r^8+4r^{11}+\cdots ,$ we get $$S(1-r^3)=r^2+r^5+r^8+r^{11}+\cdots =\frac{r^2}{1-r^3}.$$Hence, $$S=\frac{r^2}{(1-r^3{)}^2}.$$Since $r^3+\frac{2}{5}r-1=0,$ $1-r^3=\frac{2}{5}r.$ Therefore, $$S=\frac{r^2}{\frac{4}{25}{r}^2}=\boxed{\frac{25}{4}}.$$