The sum of the first $2011$ terms of a [geometric sequence](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) is $200$. The sum of the first $4022$ terms is $380$. Find the sum of the first $6033$ terms.
Since the sum of the first $2011$ terms is $200$, and the sum of the first $4022$ terms is $380$, the sum of the second $2011$ terms is $180$.
This is decreasing from the first 2011, so the common ratio is less than one.
Because it is a geometric sequence and the sum of the first 2011 terms is $200$, second $2011$ is $180$, the ratio of the second $2011$ terms to the first $2011$ terms is $\frac{9}{10}$. Following the same pattern, the sum of the third $2011$ terms is $\frac{9}{10}*180 = 162$.
Thus, $200+180+162=542$, so the sum of the first $6033$ terms is $\boxed{542}$.