1. **State the problem:** A ball bounces to a height of 3 meters initially, and each subsequent bounce reaches a height 10% lower than the previous one. We need to find the total height covered by the ball before it comes to rest. 2. **Understand the problem:** The ball goes up and then comes down for each bounce, except the very first drop is from the initial height. The height of each bounce forms a geometric sequence where each term is 90% (100% - 10%) of the previous height. 3. **Define the sequence:** Let the initial height be $h_1 = 3$ meters. The height of the second bounce is $h_2 = 3 \times 0.9 = 2.7$ meters. The height of the third bounce is $h_3 = 2.7 \times 0.9 = 3 \times 0.9^2$ meters, and so on. 4. **Total height calculation:** The ball travels up and down for each bounce except the first drop (only down). So the total height $H$ is: $$ H = h_1 + 2(h_2 + h_3 + h_4 + \cdots) = 3 + 2 \sum_{n=2}^\infty 3 \times 0.9^{n-1} $$ 5. **Simplify the sum:** Change the index to start from $n=1$ for convenience: $$ \sum_{n=2}^\infty 3 \times 0.9^{n-1} = 3 \sum_{k=1}^\infty 0.9^k $$ 6. **Sum of infinite geometric series:** For $|r|<1$, $$ \sum_{k=1}^\infty r^k = \frac{r}{1-r} $$ Here, $r=0.9$, so $$ \sum_{k=1}^\infty 0.9^k = \frac{0.9}{1-0.9} = \frac{0.9}{0.1} = 9 $$ 7. **Calculate total height:** $$ H = 3 + 2 \times 3 \times 9 = 3 + 54 = 57 $$ **Final answer:** The total height covered by the ball before coming to rest is **57 meters**.