A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to threequarters its previous height. So, after its ﬁrst bounce it rises to 75 ft, and after its second
bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?

Question

anonymous · Answer

geometry series

$$\sum_{n=0}^{\infty}75(.75)^n$$

anonymous · Answer

use formula

$$\frac{a_0}{1-r}$$

jamesj · Answer

$$ 100 + 2 \times 75 \sum_{n=0}^\infty (3/4)^n $$
First it falls 100, then it travels 75 twice, then 75(3/4) twice, then 75(3/4)^2 twice, etc.

jamesj · Answer

Make sense?  If so, what final answer do you get?

anonymous · Answer

and $ \sum \limits_{n=0}^\infty \left(\frac 3 4 \right)^n = \large \frac{  \frac3 4 }{1 - \frac3 4 }$ which is sum of infinite series, this is convergent as common ratio is less than 1.

anonymous · Answer

sum of infinite *geometric* series

jamesj · Answer

ffm, careful with the numerator.

anonymous · Answer

75, not 3/4

anonymous · Answer

oops, first term is 1; $$ \sum \limits_{n=0}^\infty \left(\frac 3 4 \right)^n = \large \frac{  1 }{1 - \frac3 4 } $$

anonymous · Answer

so the answer is 700 ft.

anonymous · Answer

Thanks. Yes I understand how it worked out. I did $$100 + \frac{2 \times 75}{1 - \frac{3}{4}}$$