1. **State the problem:** Marius earns interest each year following the sequence 200, 208, 216.30, 225, ... and we want to find the total interest earned by the 30th year. 2. **Identify the pattern:** This is a geometric sequence where each term is multiplied by a common ratio $r$ to get the next term. 3. **Find the common ratio $r$:** $$r = \frac{208}{200} = 1.04$$ 4. **General term formula for geometric sequence:** $$a_n = a_1 \times r^{n-1}$$ where $a_1 = 200$ and $r = 1.04$. 5. **Sum of the first $n$ terms of a geometric sequence:** $$S_n = a_1 \times \frac{1 - r^n}{1 - r}$$ 6. **Calculate the total interest earned by the 30th year:** $$S_{30} = 200 \times \frac{1 - 1.04^{30}}{1 - 1.04}$$ 7. **Simplify the denominator:** $$1 - 1.04 = -0.04$$ 8. **Calculate $1.04^{30}$:** $$1.04^{30} \approx 3.2434$$ 9. **Substitute values:** $$S_{30} = 200 \times \frac{1 - 3.2434}{-0.04} = 200 \times \frac{-2.2434}{-0.04}$$ 10. **Simplify the fraction:** $$\frac{-2.2434}{-0.04} = 56.085$$ 11. **Calculate the total sum:** $$S_{30} = 200 \times 56.085 = 11217$$ **Final answer:** Marius will have earned approximately **11217** in total interest by the 30th year.