Question 1169577: Pls answer my question
When a ball is tossed to a height of 4 meters above the ground, it always rebounds to 40% of it's previous height until it stops. Find the total distance that the ball has covered when it strikes the ground for the fifth time.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! each time the ball is tossed, it rebounds to 40% of the previous height.
when it is tossed, the height is 4 meters.
on the first rebound, the height is .4 * 4 = 1.6 meters.
on the second rebound, the height is .4 * 1.6 = .64 meters.
on the third rebound, the height is .4 * .64 = .256 meters.
etc.
at some point the height will decrease to 0.
this appears to be a geometric series.
the formula for the nth term of a geometric series is An = A1*r^(n-1).
the formula for the sum of the n terms of a geometric series is Sn = A1*(1-r^n)/(1-r).
if r is less than 1, than the infinite sum of the geometric series is S = A1/(1-r)
in your series, r is less than 1.
it is .4 to be exact.
the formula for the infinite sum of your series would be:
S = 4 / (1-r) = 4/.6 = 6.66666667 rounded to 8 decimal places.
to find the total distance traveled when the ball strikes the ground for the fifth time, the formula of Sn = A1*(1-r^n)/(1-r) is used.
that becomes:
S5 = 4*(1-.4^5)/.6 = 6.5984.
the ball will have traveled 6.5984 meters by the time it strikes the ground for the 5th time.
here's what the series looks like in excel.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
The solution by @Theo is incorrect, since he calculates the sums of distances in one direction at each bound,
while the correct solution should account for them twice (up and down).
So, I made the table in Excel to provide an absolute clarity.
See below.
4 m
-----------------------------
up down
1st rebound 1.6 1.6
2nd rebound 0.64 0.64
3rd rebound 0.256 0.256
4th rebound 0.1024 0.1024
5th rebound
total sum = 4 + the sum in the first column + the sum in the second column = 9.1968 meters. ANSWER
Solved.
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