First part of question:
Find the general term that represents the situation in terms of k.
The general term for geometric series is:
![a_{n}=a_{1}r^{n-1}](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7Dr%5E%7Bn-1%7D)
= the first term of the series
= the geometric ratio
would represent the height at which the ball is first dropped. Therefore:
![a_{1} = k](https://tex.z-dn.net/?f=a_%7B1%7D%20%3D%20k)
We also know that the ball has a rebound ratio of 75%, meaning that the ball only bounces 75% of its original height every time it bounces. This appears to be our geometric ratio. Therefore:
![r=\frac{3}{4}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B3%7D%7B4%7D)
Our general term would be:
![a_{n}=a_{1}r^{n-1}](https://tex.z-dn.net/?f=a_%7Bn%7D%3Da_%7B1%7Dr%5E%7Bn-1%7D)
![a_{n}=k(\frac{3}{4}) ^{n-1}](https://tex.z-dn.net/?f=a_%7Bn%7D%3Dk%28%5Cfrac%7B3%7D%7B4%7D%29%20%5E%7Bn-1%7D)
Second part of question:
If the ball dropped from a height of 235ft, determine the highest height achieved by the ball after six bounces.
represents the initial height:
![k = 235\ ft](https://tex.z-dn.net/?f=k%20%3D%20235%5C%20ft)
represents the number of times the ball bounces:
![n = 6](https://tex.z-dn.net/?f=n%20%3D%206)
Plugging this back into our general term of the geometric series:
![a_{n}=k(\frac{3}{4}) ^{n-1}](https://tex.z-dn.net/?f=a_%7Bn%7D%3Dk%28%5Cfrac%7B3%7D%7B4%7D%29%20%5E%7Bn-1%7D)
![a_{n}=235(\frac{3}{4}) ^{6-1}](https://tex.z-dn.net/?f=a_%7Bn%7D%3D235%28%5Cfrac%7B3%7D%7B4%7D%29%20%5E%7B6-1%7D)
![a_{n}=235(\frac{3}{4}) ^{5}](https://tex.z-dn.net/?f=a_%7Bn%7D%3D235%28%5Cfrac%7B3%7D%7B4%7D%29%20%5E%7B5%7D)
![a_{n}=55.8\ ft](https://tex.z-dn.net/?f=a_%7Bn%7D%3D55.8%5C%20ft)
represents the highest height of the ball after 6 bounces.
Third part of question:
If the ball dropped from a height of 235ft, find the total distance traveled by the ball when it strikes the ground for the 12th time.
This would be easier to solve if we have a general term for the <em>sum </em>of a geometric series, which is:
![S_{n}=\frac{a_{1}(1-r^{n})}{1-r}](https://tex.z-dn.net/?f=S_%7Bn%7D%3D%5Cfrac%7Ba_%7B1%7D%281-r%5E%7Bn%7D%29%7D%7B1-r%7D)
We already know these variables:
![a_{1}= k = 235\ ft](https://tex.z-dn.net/?f=a_%7B1%7D%3D%20k%20%3D%20235%5C%20ft)
![r=\frac{3}{4}](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B3%7D%7B4%7D)
![n = 12](https://tex.z-dn.net/?f=n%20%3D%2012)
Therefore:
![S_{n}=\frac{(235)(1-\frac{3}{4} ^{12})}{1-\frac{3}{4} }](https://tex.z-dn.net/?f=S_%7Bn%7D%3D%5Cfrac%7B%28235%29%281-%5Cfrac%7B3%7D%7B4%7D%20%5E%7B12%7D%29%7D%7B1-%5Cfrac%7B3%7D%7B4%7D%20%7D)
![S_{n}=\frac{(235)(1-\frac{3}{4} ^{12})}{\frac{1}{4} }](https://tex.z-dn.net/?f=S_%7Bn%7D%3D%5Cfrac%7B%28235%29%281-%5Cfrac%7B3%7D%7B4%7D%20%5E%7B12%7D%29%7D%7B%5Cfrac%7B1%7D%7B4%7D%20%7D)
![S_{n}=(4)(235)(1-\frac{3}{4} ^{12})](https://tex.z-dn.net/?f=S_%7Bn%7D%3D%284%29%28235%29%281-%5Cfrac%7B3%7D%7B4%7D%20%5E%7B12%7D%29)
![S_{n}=910.22\ ft](https://tex.z-dn.net/?f=S_%7Bn%7D%3D910.22%5C%20ft)