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8.2. GEOMETRIC SERIES
8.2 Geometric Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a geometric series?
• What is a partial sum of a geometric series? What is a simplified form of the nth
partial sum of a geometric series?
• Under what conditions does a geometric series converge? What is the sum of a
convergent geometric series?
Introduction
Many important sequences are generated through the process of addition. In Preview
Activity 8.2, we see a particular example of a special type of sequence that is connected to
a sum.
Preview Activity 8.2. Warfarin is an anticoagulant that prevents blood clotting; often it
is prescribed to stroke victims in order to help ensure blood flow. The level of warfarin
has to reach a certain concentration in the blood in order to be effective.
Suppose warfarin is taken by a particular patient in a 5 mg dose each day. The drug is
absorbed by the body and some is excreted from the system between doses. Assume that at
the end of a 24 hour period, 8% of the drug remains in the body. Let Q(n) be the amount
(in mg) of warfarin in the body before the (n + 1)st dose of the drug is administered.
(a) Explain why Q(1) = 5 × 0.08 mg.
(b) Explain why Q(2) = (5 + Q(1)) × 0.08 mg. Then show that
Q(2) = (5 × 0.08) (1 + 0.08) mg.
(c) Explain why Q(3) = (5 + Q(2)) × 0.08 mg. Then show that
Q(3) = (5 × 0.08)
1 + 0.08 + 0.08
2 mg.
(d) Explain why Q(4) = (5 + Q(3)) × 0.08 mg. Then show that
Q(4) = (5 × 0.08)
1 + 0.08 + 0.08
2 + 0.08
3 mg.
(e) There is a pattern that you should see emerging. Use this pattern to find a formula
for Q(n), where n is an arbitrary positive integer.
8.2. GEOMETRIC SERIES
8.2 Geometric Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a geometric series?
• What is a partial sum of a geometric series? What is a simplified form of the nth
partial sum of a geometric series?
• Under what conditions does a geometric series converge? What is the sum of a
convergent geometric series?
Introduction
Many important sequences are generated through the process of addition. In Preview
Activity 8.2, we see a particular example of a special type of sequence that is connected to
a sum.
Preview Activity 8.2. Warfarin is an anticoagulant that prevents blood clotting; often it
is prescribed to stroke victims in order to help ensure blood flow. The level of warfarin
has to reach a certain concentration in the blood in order to be effective.
Suppose warfarin is taken by a particular patient in a 5 mg dose each day. The drug is
absorbed by the body and some is excreted from the system between doses. Assume that at
the end of a 24 hour period, 8% of the drug remains in the body. Let Q(n) be the amount
(in mg) of warfarin in the body before the (n + 1)st dose of the drug is administered.
(a) Explain why Q(1) = 5 × 0.08 mg.
(b) Explain why Q(2) = (5 + Q(1)) × 0.08 mg. Then show that
Q(2) = (5 × 0.08) (1 + 0.08) mg.
(c) Explain why Q(3) = (5 + Q(2)) × 0.08 mg. Then show that
Q(3) = (5 × 0.08)
1 + 0.08 + 0.08
2 mg.
(d) Explain why Q(4) = (5 + Q(3)) × 0.08 mg. Then show that
Q(4) = (5 × 0.08)
1 + 0.08 + 0.08
2 + 0.08
3 mg.
(e) There is a pattern that you should see emerging. Use this pattern to find a formula
for Q(n), where n is an arbitrary positive integer.