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.
