# Thread: [SOLVED] Geometric Series Word Problem

1. ## [SOLVED] Geometric Series Word Problem

I now how to workout this problem, but I make a few small errors along the way and I need some sort of checking or help.

Problem.

A ball is drooped from a height of 10 feet (Assume no air resistance), each bounce is $(\frac{3}{4})$of the height of the bounce before. Thus, after the balls hit the floor for the first time, the ball rises to a height of $10(\frac{3}{4})$, after the second time, $10(\frac{3}{4})^2$.

A. Find an expression for the height which the ball rises
Problem. after it hits the floor for the nth time.

$\sum_{n=0}^{\infty} 10\cdot\frac{3}{4}^n$

Thus, $H_0 = 10$, $H_1 = 10\frac{3}{4}$, $H_2 = 10(\frac{3}{4})^2$ $H_n = 10(\frac{3}{4})^n$

B. Find an expression for the total vertical distance the ball has travled for the first, second, third, and fourth times.

$D_n=H_0+2H_1+2H_3.......+2H_{n-1}$

$D_1=10+2\cdot10(\frac{3}{4})$

$D_2=10+2\cdot10(\frac{3}{4})+2\cdot10(\frac{3}{4}) ^2$

$D_3=10+2\cdot10(\frac{3}{4})+2\cdot10(\frac{3}{4}) ^2+2\cdot10(\frac{3}{4})^3$

$D_3=10+2\cdot10(\frac{3}{4})+2\cdot10(\frac{3}{4}) ^2+2\cdot10(\frac{3}{4})^3+2\cdot10(\frac{3}{4})^4$

D. Find the closed form expression for the total distance traveled for the nth time, this the part where I get my confused in properly making a simplify formula and simplifying it

2. Why sum from 0 to infinity? Wait, actually, why sum at all?!

after it's first dropped, the ball will rise to a height of $10 \cdot \frac{3}{4}$. The second time, it will rise to a height of $10 \cdot \left( \frac{3}{4} \right)^2$, and so on..

So the nth rise is the nth term of the geometric series with $a_0 = 10, \ q = \frac{3}{4}$, which is..?

3. HI

$10+2(7.5)+2(5.625)+...$

Do you see why it must be multiplied by 2 ? Because we have to consider both the rebound and the path taken to fall again .

$10+2[7.5+5.625+...]$

$10+2\sum^{n}_{r=1}10\cdot (\frac{3}{4})^r$

$
10+20\sum^{n}_{r=1}(\frac{3}{4})^r
$

4. Originally Posted by Defunkt
Why sum from 0 to infinity? Wait, actually, why sum at all?!

after it's first dropped, the ball will rise to a height of $10 \cdot \frac{3}{4}$. The second time, it will rise to a height of $10 \cdot \left( \frac{3}{4} \right)^2$, and so on..

So the nth rise is the nth term of the geometric series with $a_0 = 10, \ q = \frac{3}{4}$, which is..?
Defunkt, I was asking the same thing till I recongized, they are asking for total distance, when H_0 = 10 is the measurement the ball had to drop to the floor that what got my constants messed up when working with the closed form

Thanks mathaddict I was looking for the correct constants to pull out, Ill try re-working the problem with your advice

5. Ehhh, this is a tricky little son of a gun, they pulled the first H_1 out and up the summation as such

$10+20(\frac{3}{4})\sum_{n=1}^n(\frac{3}{4})^{n-1}$

However, I was able to match for a power of n