8.2. GEOMETRIC SERIES
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(c) Find a formula for the amount paid on day n, as well as for the total amount
paid by day n. Use this formula to determine which option (1 or 2) you should
take.
2. Suppose you drop a golf ball onto a hard surface from a height h. The collision with
the ground causes the ball to lose energy and so it will not bounce back to its original
height. The ball will then fall again to the ground, bounce back up, and continue.
Assume that at each bounce the ball rises back to a height
3
4 of the height from which
it dropped. Let h n be the height of the ball on the nth bounce, with h 0 = h. In this
exercise we will determine the distance traveled by the ball and the time it takes to
travel that distance.
(a) Determine a formula for h 1 in terms of h.
(b) Determine a formula for h 2 in terms of h.
(c) Determine a formula for h 3 in terms of h.
(d) Determine a formula for h n in terms of h.
(e) Write an infinite series that represents the total distance traveled by the ball.
Then determine the sum of this series.
(f) Next, let’s determine the total amount of time the ball is in the air.
(i) When the ball is dropped from a height H, if we assume the only force
acting on it is the acceleration due to gravity, then the height of the ball at
time t is given by
H −
1
2
gt
2 .
Use this formula to determine the time it takes for the ball to hit the
ground after being dropped from height H.
(ii) Use your work in the preceding item, along with that in (a)-(e) above to
determine the total amount of time the ball is in the air.
3. Suppose you play a game with a friend that involves rolling a standard six-sided die.
Before a player can participate in the game, he or she must roll a six with the die.
Assume that you roll first and that you and your friend take alternate rolls. In this
exercise we will determine the probability that you roll the first six.
(a) Explain why the probability of rolling a six on any single roll (including your
first turn) is
1
6 .
(b) If you don’t roll a six on your first turn, then in order for you to roll the first
six on your second turn, both you and your friend had to fail to roll a six on
your first turns, and then you had to succeed in rolling a six on your second
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(c) Find a formula for the amount paid on day n, as well as for the total amount
paid by day n. Use this formula to determine which option (1 or 2) you should
take.
2. Suppose you drop a golf ball onto a hard surface from a height h. The collision with
the ground causes the ball to lose energy and so it will not bounce back to its original
height. The ball will then fall again to the ground, bounce back up, and continue.
Assume that at each bounce the ball rises back to a height
3
4 of the height from which
it dropped. Let h n be the height of the ball on the nth bounce, with h 0 = h. In this
exercise we will determine the distance traveled by the ball and the time it takes to
travel that distance.
(a) Determine a formula for h 1 in terms of h.
(b) Determine a formula for h 2 in terms of h.
(c) Determine a formula for h 3 in terms of h.
(d) Determine a formula for h n in terms of h.
(e) Write an infinite series that represents the total distance traveled by the ball.
Then determine the sum of this series.
(f) Next, let’s determine the total amount of time the ball is in the air.
(i) When the ball is dropped from a height H, if we assume the only force
acting on it is the acceleration due to gravity, then the height of the ball at
time t is given by
H −
1
2
gt
2 .
Use this formula to determine the time it takes for the ball to hit the
ground after being dropped from height H.
(ii) Use your work in the preceding item, along with that in (a)-(e) above to
determine the total amount of time the ball is in the air.
3. Suppose you play a game with a friend that involves rolling a standard six-sided die.
Before a player can participate in the game, he or she must roll a six with the die.
Assume that you roll first and that you and your friend take alternate rolls. In this
exercise we will determine the probability that you roll the first six.
(a) Explain why the probability of rolling a six on any single roll (including your
first turn) is
1
6 .
(b) If you don’t roll a six on your first turn, then in order for you to roll the first
six on your second turn, both you and your friend had to fail to roll a six on
your first turns, and then you had to succeed in rolling a six on your second