1. College Algebra Fifth Edition
James Stewart  Lothar Redlin  Saleem Watson

2. Sequences and Series 9

3. 9.3
Geometric Sequences

4. Introduction
Geometric sequences occur frequently in applications to fields such as:
Finance
Population growth

5. Geometric Sequences

6. Arithmetic Sequence vs. Geometric Sequence
An arithmetic sequence is generated when:
We repeatedly add a number d to an initial term a.
A geometric sequence is generated when:
We start with a number a and repeatedly multiply by a fixed nonzero constant r.

7. Geometric Sequence—Definition
A geometric sequence is a sequence of the form
a, ar, ar2, ar3, ar4, . . .
The number a is the first term.
r is the common ratio of the sequence.
The nth term of a geometric sequence is given by: an = arn–1

8. The number r is called the common ratio because:
The ratio of any two consecutive terms of the sequence is r.

9. E.g. 1—Geometric Sequences
Example (a)
If a = 3 and r = 2, then we have the geometric sequence
3, 3 · 2, 3 · 22, 3 · 23, 3 · 24,
or , 6, 12, 24, 48, . . .
Notice that the ratio of any two consecutive terms is r = 2.
The nth term is: an = 3(2)n–1

10. E.g. 1—Geometric Sequences
Example (b)
The sequence
2, –10, 50, –250, 1250, . . .
is a geometric sequence with a = 2 and r = –5
When r is negative, the terms of the sequence alternate in sign.
The nth term is: an = 2(–5)n–1.

11. E.g. 1—Geometric Sequences
Example (c)
The sequence
is a geometric sequence with a = 1 and r = ⅓
The nth term is: an = (⅓)n–1

12. E.g. 1—Geometric Sequences
Example (d)
Here’s the graph of the geometric sequence an = (1/5) · 2n – 1
Notice that the points in the graph lie on the graph of the exponential function y = (1/5) · 2x–1

13. E.g. 1—Geometric Sequences
Example (d)
If 0 < r < 1, then the terms of the geometric sequence arn–1 decrease.
However, if r > 1, then the terms increase.
What happens if r = 1?

14. Geometric Sequences in Nature
Geometric sequences occur naturally.
Here is a simple example.
Suppose a ball has elasticity such that, when it is dropped, it bounces up one-third of the distance it has fallen.

15. Geometric Sequences in Nature
If the ball is dropped from a height of 2 m, it bounces up to a height of 2(⅓) = ⅔ m.
On its second bounce, it returns to a height of (⅔)(⅓) = (2/9)m, and so on.

16. Geometric Sequences in Nature
Thus, the height hn that the ball reaches on its nth bounce is given by the geometric sequence
hn = ⅔(⅓)n–1 = 2(⅓)n
We can find the nth term of a geometric sequence if we know any two terms—as the following examples show.

17. E.g. 2—Finding Terms of a Geometric Sequence
Find the eighth term of the geometric sequence 5, 15, 45,
To find a formula for the nth term of this sequence, we need to find a and r.
Clearly, a = 5.

18. E.g. 2—Finding Terms of a Geometric Sequence
To find r, we find the ratio of any two consecutive terms.
For instance, r = (45/15) = 3
Thus, an = 5(3)n–1
The eighth term is: a8 = 5(3)8–1 = 5(3)7 = 10,935

19. E.g. 3—Finding Terms of a Geometric Sequence
The third term of a geometric sequence is 63/4, and the sixth term is 1701/32.
Find the fifth term.
Since this sequence is geometric, its nth term is given by the formula an = arn–1.
Thus, a3 = ar3–1 = ar2 a6 = ar6–1 = ar5

20. E.g. 3—Finding Terms of a Geometric Sequence
From the values we are given for those two terms, we get this system of equations:
We solve this by dividing:

21. E.g. 3—Finding Terms of a Geometric Sequence
Substituting for r in the first equation, 63/4 = ar2, gives:
It follows that the nth term of this sequence is: an = 7(3/2)n–1
Thus, the fifth term is:

22. Partial Sums of Geometric Sequences

23. Partial Sums of Geometric Sequences
For the geometric sequence a, ar, ar2, ar3, ar4, , arn–1, , the nth partial sum is:

24. Partial Sums of Geometric Sequences
To find a formula for Sn, we multiply Sn by r and subtract from Sn:

25. Partial Sums of Geometric Sequences
We summarize this result as follows.

26. Partial Sums of a Geometric Sequence
For the geometric sequence an = arn–1, the nth partial sum Sn = a + ar + ar2 + ar3 + ar arn– (r ≠ 1) is given by:

27. E.g. 4—Finding a Partial Sum of a Geometric Sequence
Find the sum of the first five terms of the geometric sequence 1, 0.7, 0.49, 0.343, . . .
The required sum is the sum of the first five terms of a geometric sequence with a = 1 and r = 0.7

28. E.g. 4—Finding a Partial Sum of a Geometric Sequence
Using the formula for Sn with n = 5, we get:
The sum of the first five terms of the sequence is

29. E.g. 5—Finding a Partial Sum of a Geometric Sequence
Find the sum
The sum is the fifth partial sum of a geometric sequence with first term a = 7(–⅔) = –14/3 and common ratio r = –⅔.
Thus, by the formula for Sn, we have:

30. What Is an Infinite Series?

31. Infinite Series
An expression of the form a1 + a2 + a3 + a is called an infinite series.
The dots mean that we are to continue the addition indefinitely.

32. What meaning can we attach to the sum of infinitely many numbers?
Infinite Series
What meaning can we attach to the sum of infinitely many numbers?
It seems at first that it is not possible to add infinitely many numbers and arrive at a finite number.
However, consider the following problem.

33. You have a cake and you want to eat it by:
Infinite Series
You have a cake and you want to eat it by:
First eating half the cake.
Then eating half of what remains.
Then again eating half of what remains.

34. Infinite Series
This process can continue indefinitely because, at each stage, some of the cake remains.
Does this mean that it’s impossible to eat all of the cake?
Of course not.

35. Let’s write down what you have eaten from this cake:
Infinite Series
Let’s write down what you have eaten from this cake:
This is an infinite series.
We note two things about it.

36. The more terms of this series we add, the closer the sum is to 1.
Infinite Series
It’s clear that, no matter how many terms of this series we add, the total will never exceed 1.
The more terms of this series we add, the closer the sum is to 1.

37. Infinite Series
This suggests that the number 1 can be written as the sum of infinitely many smaller numbers:

38. Infinite Series
To make this more precise, let’s look at the partial sums of this series:

39. In general (see Example 5 of Section 9.1),
Infinite Series
In general (see Example 5 of Section 9.1),
As n gets larger and larger, we are adding more and more of the terms of this series.
Intuitively, as n gets larger, Sn gets closer to the sum of the series.

40. Now, notice that, as n gets large, (1/2n) gets closer and closer to 0.
Infinite Series
Now, notice that, as n gets large, (1/2n) gets closer and closer to 0.
Thus, Sn gets close to 1 – 0 = 1.
Using the notation of Section 4.6, we can write:
Sn → 1 as n → ∞

41. Sum of Infinite Series
In general, if Sn gets close to a finite number S as n gets large, we say that:
S is the sum of the infinite series.

42. Infinite Geometric Series

43. Infinite Geometric Series
An infinite geometric series is a series of the form
a + ar + ar2 + ar3 + ar arn–
We can apply the reasoning used earlier to find the sum of an infinite geometric series.

44. Infinite Geometric Series
The nth partial sum of such a series is given by:
It can be shown that, if | r | < 1, rn gets close to 0 as n gets large.
You can easily convince yourself of this using a calculator.

45. Infinite Geometric Series
It follows that Sn gets close to a/(1 – r) as n gets large, or
Thus, the sum of this infinite geometric series is: a/(1 – r)

46. Sum of an Infinite Geometric Series
If | r | < 1, the infinite geometric series a + ar + ar2 + ar3 + ar arn– has the sum

47. E.g. 6—Finding the Sum of an Infinite Geometric Series
Find the sum of the infinite geometric series
We use the formula.
Here, a = 2 and r = (1/5).
So, the sum of this infinite series is:

48. E.g. 7—Writing a Repeated Decimal as a Fraction
Find the fraction that represents the rational number
This repeating decimal can be written as a series:

49. E.g. 7—Writing a Repeated Decimal as a Fraction
After the first term, the terms of the series form an infinite geometric series with:

50. E.g. 7—Writing a Repeated Decimal as a Fraction
So, the sum of this part of the series is:
Thus,