1. ## geometric series

Can someone tell me what im doing worng

A ball drops from a height of 24 feet. Each time it hits the ground, it bounces up 30 percents of the height it fall. Assume it goes on forever, find the total distance it travels.

This is what I've done Sum from n=1 to infinity of (3/10)^n times 24
= 24/(1-3/10) = 240/7.

2. Hello,

Before the first bounce, it travelled on 24 feet... So I think you should put it in the solution

3. i see what i did wrong, it was a dumb mistake, thanks

4. Hello, matt90!

This is a classic trick question . . .

A ball drops from a height of 24 feet.
Each time it hits the ground, it bounces up 30% of the height it fell.
Assume it goes on forever, find the total distance it travels.
Here's the "trick" part of it . . .

The ball falls 24 feet.

It bounces up $30\% \times 24 = 8$ feet . . . Then it falls 8 feet.

Then it bounces up $30\% \times 8 = 2.4$ feet . . . Then it falls 2.4 feet.

. . and so on . . .

The total distance is: . $S \;=\;24 + {\color{red}2}(24)(0.3) + {\color{red}2}(24)(0.3)^2 + {\color{red}2}(24)(0.3)^3 + \hdots$

$\text{We have: }\;S \;=\;24 + 2(24)(0.3)\underbrace{\bigg[1 + 0.3 + 0.3^2 + 0.3^3 + \hdots\bigg]}_{\text{geometric series}}$

The sum of this geometric series is: . $\frac{1}{1-\frac{3}{10}} \:=\:\frac{10}{7}$

Therefore: . $S \;=\;24 + 14.4\left(\frac{10}{7}\right) \;=\;\frac{312}{7} \;=\;\boxed{44\frac{4}{7}\text{ feet}} \$