A ball is dropped from $405$ meters and rebounds two-thirds the distance it falls each time it bounces. How many meters will the ball have traveled when it hits the ground the fourth time?
I need to figure out how far the ball travels each time it bounces, and then add up those distances to get the total. The first time it hits the ground, it travels $405$ meters. Then it rebounds two-thirds of that, which is $270$ meters, and falls back the same distance. So the second time it hits the ground, it has traveled $405 + 270 + 270 = 945$ meters. Then it rebounds two-thirds of $270$, which is $180$ meters, and falls back the same distance. So the third time it hits the ground, it has traveled $945 + 180 + 180 = 1305$ meters. Then it rebounds two-thirds of $180$, which is $120$ meters, and falls back the same distance. So the fourth time it hits the ground, it has traveled $1305 + 120 + 120 = 1545$ meters. 