A ball is dropped from a height of 20 ft and always rebounds 1/4 of the distance fallen.
A. write a sequence for the first 3 rebound heights
B. How high is the 6th rebound?
C. Find the sum of all possible rebounds.
I keep getting numbers that are too high. I don't think I have the correct geometric sequence.
1st rebound 20*1/4
2nd 20*(1/4)^2
3rd 20*(1/4)^3
6th 20*(1/4)^6
The sum=20*(1/4+(1/4)^2+(1/4)^3+...)=
20*(1/4)/(1-1/4)=20/3
To find the correct geometric sequence for the rebounds, we need to understand the given information.
Let's call the height of the ball after each rebound "h".
Given that the ball always rebounds 1/4 of the distance fallen, we can express the height after each rebound as:
First rebound: h = 20 * (1/4)
Second rebound: h = (20 * (1/4)) * (1/4) = 20 * (1/4)^2
Third rebound: h = (20 * (1/4)) * (1/4)^2 * (1/4) = 20 * (1/4)^3
A. Now, let's calculate the first 3 rebound heights using the formula:
First rebound height: h1 = 20 * (1/4) = 5 ft
Second rebound height: h2 = 20 * (1/4)^2 = 1.25 ft
Third rebound height: h3 = 20 * (1/4)^3 = 0.3125 ft
B. To find the height of the 6th rebound, we can plug in the values into the formula:
h6 = 20 * (1/4)^6 = 0.03125 ft
C. To find the sum of all possible rebounds, we can use the formula for the sum of a geometric series. However, in this case, since the ball rebounds indefinitely, we can use the formula for the sum of an infinite geometric series.
The sum of an infinite geometric series is given by the formula:
Sum = a / (1 - r)
In this case, the first term (a) is 20 * (1/4) = 5 ft, and the common ratio (r) is 1/4.
Sum = 5 / (1 - 1/4)
Sum = 5 / (3/4)
Sum = 5 * 4/3
Sum = 20/3 ft
So, the sum of all possible rebounds is 20/3 ft.
Make sure you double-check your calculations and use the correct formula to ensure accurate results.