Question 1200022: A ball that bounces up half the distance from which it is dropped is released one metre from the floor.If it continues bouncing forever what is the total distance it travels?
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! For the 1st bounce there is a one-way trip from the height of 1 m. For each subsequent bounce the ball starts out and returns to the same height. Each
successive bounce is 1/2 the previous bounce, which suggests we have a geometric
progression with common ratio of 1/2. Since every bounce but the 1st the distance
is twice the height each term will be multiplied by two.
The first few distances are:
1,1,1/2,1/4,1/8, ...
If the n-th term of our progression is a_n = 2*(1/2)^(n-1), this gives
n=1: 2
n=2: 1
n=3: 1/2
n=4: 1/4
We see that this matches the distances if we subtract 1 from the first term.
The sum of an infinite progression S_n = a1/(1-r) = 2/(1-1/2) = 4
Thus, the total distance is S_n - 1 = 4 - 1 = 3.
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