A certain ball is dropped from an altitude of 1 meter and rebounds one-half of it's previous altitude. What is the total distance the ball has traveled when it hits the floor the 40th time?

To find the total distance the ball has traveled when it hits the floor the 40th time, we need to sum up the distances it travels on each bounce.

We can start by finding the distance traveled on the first bounce. The ball is dropped from an altitude of 1 meter, so on the first bounce, it reaches an altitude of (1/2) meters.

The distance traveled on the first bounce is the sum of the descending and ascending distances. The descending distance is 1 meter, and the ascending distance is (1/2) meters. Therefore, on the first bounce, the ball travels a total distance of 1 meter + (1/2) meter = (3/2) meters.

For subsequent bounces, we can see a pattern. On each bounce, the ball reaches an altitude that is half of the previous bounce. This means that the descending distance remains the same (1 meter), but the ascending distance decreases by half.

To simplify the calculation, let's look at the distances in terms of a geometric series. The first term is 1 meter, and the common ratio is 1/2. We need to find the sum of the first 40 terms of this series.

To calculate the sum of a geometric series, we can use the formula:

S = a * (1 - r^n) / (1 - r)

Where:
S is the sum of the series
a is the first term of the series
r is the common ratio
n is the number of terms in the series

Plugging in the values, we have:
S = 1 * (1 - (1/2)^40) / (1 - 1/2)

Evaluating this expression will give us the total distance traveled by the ball when it hits the floor the 40th time.