1. Problem 12: Find the sum to infinity of the series $$3 - 2 + \frac{4}{3} - \frac{8}{9} + ...$$ 2. Identify the pattern in the series: - The first term $$a_1 = 3$$ - The second term is $$-2$$ - The third term is $$\frac{4}{3}$$ - The fourth term is $$-\frac{8}{9}$$ 3. Check if the series is geometric: Calculate the ratio $$r$$ between consecutive terms: $$r = \frac{-2}{3} = -\frac{2}{3}$$ Check next ratio: $$\frac{\frac{4}{3}}{-2} = -\frac{2}{3}$$ Thus, $$r = -\frac{2}{3}$$ 4. The series is geometric with first term $$a = 3$$ and common ratio $$r = -\frac{2}{3}$$. 5. The sum to infinity of a geometric series with $$|r| < 1$$ is $$S_\infty = \frac{a}{1-r}$$ 6. Substitute values: $$S_\infty = \frac{3}{1 - (-\frac{2}{3})} = \frac{3}{1 + \frac{2}{3}} = \frac{3}{\frac{5}{3}} = 3 \times \frac{3}{5} = \frac{9}{5}$$ 7. Final answer for problem 12 is $$\boxed{\frac{9}{5}}$$ --- 8. Problem 13: A ball is dropped from a height of 6 m. Each time it hits the ground, it bounces up to half the previous height. Find the total distance traveled until it stops. 9. The ball travels down 6 m initially. 10. After hitting the ground, it bounces up $$\frac{6}{2} = 3$$ m, then falls 3 m. 11. Next bounce up is $$\frac{3}{2} = 1.5$$ m, then falls 1.5 m, and so on. 12. Total distance $$D$$ equals the initial fall plus twice the sum of all bounce heights: $$D = 6 + 2 \left(3 + 1.5 + 0.75 + ... \right)$$ 13. The bounce heights form a geometric series with first term $$3$$ and ratio $$r = \frac{1}{2}$$. 14. Sum of infinite bounce heights: $$S = \frac{3}{1 - \frac{1}{2}} = \frac{3}{\frac{1}{2}} = 6$$ 15. Total distance: $$D = 6 + 2 \times 6 = 6 + 12 = 18$$ 16. Final answer for problem 13 is $$\boxed{18}$$ meters.