1. **Problem Statement:** Calculate the total overhead costs for a job estimated to take 12 days, where the overhead costs increase each day in a geometric progression starting with 5000 on day 1, 7500 on day 2, 11250 on day 3, and so on. 2. **Identify the pattern:** The costs form a geometric sequence where the first term $a = 5000$ and the common ratio $r = \frac{7500}{5000} = 1.5$. 3. **Formula for the sum of the first $n$ terms of a geometric series:** $$ S_n = a \frac{r^n - 1}{r - 1} $$ 4. **Apply the formula:** For $n = 12$ days, $$ S_{12} = 5000 \times \frac{1.5^{12} - 1}{1.5 - 1} $$ 5. **Calculate $1.5^{12}$:** $$ 1.5^{12} \approx 129.7463379 $$ 6. **Calculate the sum:** $$ S_{12} = 5000 \times \frac{129.7463379 - 1}{0.5} = 5000 \times \frac{128.7463379}{0.5} = 5000 \times 257.4926758 = 1,287,463.379 $$ 7. **Interpretation:** The total overhead cost over 12 days is approximately 1,287,463.38. **Final answer:** $$\boxed{1,287,463.38}$$