1. **Problem statement:** Evaluate the sum $\sum_{r=1}^9 3^r$ to an appropriate degree of accuracy. 2. **Formula used:** This is a geometric series with first term $a = 3^1 = 3$ and common ratio $r = 3$. The sum of the first $n$ terms of a geometric series is given by: $$ S_n = a \frac{r^n - 1}{r - 1} $$ 3. **Apply the formula:** $$ S_9 = 3 \frac{3^9 - 1}{3 - 1} $$ 4. **Calculate powers and simplify:** $$ 3^9 = 19683 $$ So, $$ S_9 = 3 \frac{19683 - 1}{2} = 3 \frac{19682}{2} $$ 5. **Simplify the fraction:** $$ S_9 = 3 \times 9841 = 29523 $$ 6. **Final answer:** $$ \boxed{29523} $$ This is the sum of $3^r$ from $r=1$ to $9$.