jmc

Let the first term be $a,$ and let the common ratio be $r.$ Then $$a+ar+a{r}^2+\cdots +a{r}^{2010}=200$$and $$a+ar+a{r}^2+\cdots +a{r}^{4021}=380.$$Subtracting these equations, we get $$a{r}^{2011}+a{r}^{2012}+\cdots +a{r}^{4021}=180.$$Then $$r^{2011}(a+ar+\cdots +a{r}^{2010})=180,$$so $$r^{2011}=\frac{180}{200}=\frac{9}{10}.$$Then the sum of the first 6033 terms is $$\begin{array}{rl}a+ar+a{r}^2+\cdots +a{r}^{6032}& =(a+ar+a{r}^2+\cdots +a{r}^{4021})+(a{r}^{4022}+a{r}^{4023}+\cdots +a{r}^{6032})\\ & =380+r^{4022}(a+ar+\cdots +a{r}^{2010})\\ & =380+{\left(\frac{9}{10}\right)}^2\cdot 200\\ & =\boxed{542}.\end{array}$$