Sequence problem in calculus 2 Here is the description: Sally is playing with her bouncy ball. She starts off by dropping the ball from 10 feet above the ground. Each time her ball falls, it only bounces back up to 50% of its height.


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*How far has the ball traveled when it hits the ground the 1st time, 2nd? 3rd?

*If the ball continues to bounce indefinitely, what's the total distance traveled by the ball?


My thoughts:


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1. 


First time: 10ft (initialFall) 
Second time: 10ft(initialFall) + 5ft(bounce) + 5ft(fall)
3rd time: 10ft(initialFall) + 5ft(bounce) +5ft(fall) + 2.5ft(bounce) + 2.5ft(fall)
(Does this make sense?)

2. 


If I need to take a limit here, how do I do that?
 A: As you may have noticed, the ball only jumps to half the height of its last bounce. 
Let us call $h_i $ the height the ball reaches after bouncing $i $ times. We see that $h_{i+1} = 0.5h_i $ and $h_1 = 5$.  Also, everytime the ball bounces up, it travels $h_i $ feet going up and then $h_i $ feet going down. So we have to take "both travels" into account when calculating the total distance traveled.
We can now set $d_i$ as the distance traveled up to when the ball is going to make its ith bounce. That is, $d_i $ is the distance traveled when tha ball fell 10ft, plus the up and down of the first bounce, the second, ..., all up until the moment when the ball is touching the ground for the ith time. (Note that this $d_i $ matches your calculations for the first fall, the second fall, the third when i=1,2,3. It is just a generalization of that, but for any number of falls!)
Then $d_i = 10 + \sum_{k=1}^{i-1} h_k $. All you have to do now is calculate $\sum_{k=1}^{i-1} h_k $ when $i $ approaches infinity using $h_{i+1} = 0.5h_i $ and the formula for the geometric series.
Start by showing that $h_i = (0.5^{i-1})h_1$.
You then get $d_i = 10 + \sum_{k=1}^{i-1} (0.5^{k-1})h_1 = 10 + (0.5^{-1})\sum_{k=1}^{i-1} (0.5^{k})h_1 $. If you want the total distance traveled, take the limit 
$$\lim_{i\rightarrow\infty} d_i = \lim_{i\rightarrow\infty}\ (0.5^{-1})\sum_{k=1}^{i-1} (0.5^{k})h_1 $$ and the summation should strike you as a geometric series. Can you take it from here?
A: Total distance travelled: 
$$\begin{align}
&\;\;\;\;\;10+2(5+2.5+\cdots)\\
&=-10+2(10+5+2.5+\cdots)\\
&=-10+2(10)(1+0.5+0.5^2+0.5^3+\cdots)\\
&=-10+2(10)\frac 1{1-0.5}\\
&=30\end{align}$$
