Question

I had to drop a ball from 15ft, and i recorded the height of each bounce.
15ft
13ft
11ft
9ft
7ft
How do i use the geometric series formula to find the total distance of the height the ball has traveled?

Answer 1

The sum of a geometric series for |r| < 1 is given by
\[\sum_{n=0}^{\infty}a _{1}*r ^{n} = \frac{ a _{1} }{ 1-r }\]

Answer 2

how do i know what the r^n and the a_1 represent?

Answer 3

Are you sure this is a geometric series? It has to differ by something constant: For example if your first term was 16 and r = 1/2, then your series would be
16 + 8 + 4 + 2 + 1 + 1/2 +...

Here your terms differ by a difference of 2. That's not geometric.

Answer 4

@what do i do then? bc thats what im being asked

Answer 5

@khoala4pham

Answer 6

is there a way to turn this into a geometric series?

Answer 7

I don't think so. A geometric series must have each next term multiplied by some fractional constant (if it converges)

Answer 8

how can i switch up the numbers a little to make it work?

Answer 9

@khoala4pham

Answer 10

I don't know honestly. It's an arithmetic sequence not a geometric. As of yet, I can't think of any transformations that will make it converge.

Answer 11

can you come up with a completely different heights to make it work? i just made these up. i didnt have to drop it from 15 ft

Answer 12

Are you positive these are the numbers? The way I see it is that if you continue the pattern, it will be 9ft, 7ft, 5ft, 3ft, 1ft...what's next. Physically the ball bounces "infinitely" many times but it will converge. The ball bounces in the path of a geometric series yet this is arithetic.

Answer 13

Wait, you made these numbers up? So what are the real numbers?

Answer 14

i was supposed to come up with numbers.i wasnt given any numbers

Answer 15

OH. Okay. Let's just make up some numbers then! That sounds fun. xD
So let's say you started with a height of 16 and after each bounce, the ball bounces only 3/4 the height of the previous:
16, 12, 9...etc...
if you look at the formula above, you see that a1 is 16 and r = 3/4.
Thus the sum that the ball travels is 16/(1-3/4) = 16/(0.25) = 64. Now that is an impressive number.

Answer 16

wait, so after 9 its 7? and after that its 6? howd you get 3/4?

Answer 17

i need 5 heights, i dont get this

Answer 18

No, I'm making up an entirely new series. A geeomtric series differs in each of its terms by a constant. For example, if I decided that each term would be half the previous starting with 32, my series would be
32 + 16 + 8 + 4 + 2 + 1 + 1/2 + 1/4 +...
And this pattern continues indefinitely.
However, the sum, the infinite sum, DOES exist. Its value is 64. Because a1 = 32 and r = 1/2. Thus S = a1/(1-r) = 32/ (0.5) = 64.

Answer 19

In my problem above, I set that each next one will be 3/4 the height of the previous, starting with 16:
so our series is
16 + 12 + 9 + 27/4 + 81/16 +...
Do not be alerted by the sudden increase in the numbers such as 27 and 81--they are actually really small values.

Answer 20

how would i come up with 2 other different series that also start off with a height of 16?

Answer 21

Change the rate at which it falls. For example, another serie could be that the original height is 16 but the ball only bounces back 1/2 the previous height. Then you could try when the ball is really bad and bounces back 1/4 the previous height. (I'm choosing these numbers because they are really easy to multiply and divide with 16.

Answer 22

oh, i get it. thanks sooo much. i´ve been trying to get help on this assignment for days now