This document discusses geometric sequences and series. It begins by defining key terms like geometric sequence, common ratio, and geometric mean. Examples are provided to show how to determine if a sequence is geometric, find subsequent terms using the common ratio, and calculate geometric means and sums of geometric series. The document aims to teach students how to work with geometric sequences and series.

1. UNIT 12.3/12.5 GEOMETRICUNIT 12.3/12.5 GEOMETRIC
SEQUENCES AND SERIESSEQUENCES AND SERIES

2. Warm Up
Simplify.
1. 2.
3. (–2)8
4.
Solve for x.
5.
Evaluate.
96
256

3. Find terms of a geometric sequence,
including geometric means.
Find the sums of geometric series.
Objectives

4. geometric sequence
geometric mean
geometric series
Vocabulary

5. Serena Williams was the winner out of 128 players
who began the 2003 Wimbledon Ladies’ Singles
Championship. After each match, the winner
continues to the next round and the loser is
eliminated from the tournament. This means that
after each round only half of the players remain.

6. The number of players remaining after each round
can be modeled by a geometric sequence. In a
geometric sequence, the ratio of successive
terms is a constant called the common ratio
r (r ≠ 1) . For the players remaining, r is .

7. Recall that exponential
functions have a common
ratio. When you graph the
ordered pairs (n, an) of a
geometric sequence, the
points lie on an exponential
curve as shown. Thus, you
can think of a geometric
sequence as an exponential
function with sequential
natural numbers as the
domain.

8. Check It Out! Example 1a
Determine whether the sequence could be
geometric or arithmetic. If possible, find the
common ratio or difference.
Differences
It could be geometric with
Ratios

9. 1.7, 1.3, 0.9, 0.5, . . .
Check It Out! Example 1b
Determine whether the sequence could be
geometric or arithmetic. If possible, find the
common ratio or difference.
1.7 1.3 0.9 0.5
Differences –0.4 –0.4 –0.4
It could be arithmetic, with r = –0.4.
Ratio

10. Check It Out! Example 1c
Determine whether each sequence could be
geometric or arithmetic. If possible, find the
common ratio or difference.
–50, –32, –18, –8, . . .
–50, –32, –18, –8, . . .
Differences 18 14 10
It is neither.
Ratios

11. Each term in a geometric sequence is the product of
the previous term and the common ratio, giving the
recursive rule for a geometric sequence.
an = an–1rnth term
Common
ratio
First term

12. You can also use an explicit rule to find the nth term
of a geometric sequence. Each term is the product
of the first term and a power of the common ratio as
shown in the table.
This pattern can be generalized into a rule for all
geometric sequences.

14. Check It Out! Example 2a
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.

15. Check It Out! Example 2a Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r n–1
General rule
The 9th term is .
Substitute for a1, 9 for
n, and for r.

16. Check It Out! Example 2a Continued
Check Extend the sequence.
Given
a6 =
a7 =
a8 =
a9 =

17. 0.001, 0.01, 0.1, 1, 10, . . .
Check It Out! Example 2b
Find the 9th term of the geometric sequence.
Step 1 Find the common ratio.

18. Check It Out! Example 2b Continued
Step 2 Write a rule, and evaluate for n = 9.
an = a1 r n–1
a9 = 0.001(10)9–1
= 0.001(100,000,000) = 100,000
The 7th term is 100,000.
General rule
Substitute 0.001 for a1,
9 for n, and 10 for r.

19. Check It Out! Example 2b Continued
Check Extend the sequence.
a6 = 10(10) = 100
a7 = 100(10) = 1,000
a8 = 1,000(10) = 10,000
a9 = 10,000(10) = 100,000
Givena5 = 10

20. When given two terms of a sequence, be
sure to consider positive and negative
values for r when necessary.
Caution!

21. Check It Out! Example 3a
Find the 7th term of the geometric sequence
with the given terms.
a4 = –8 and a5 = –40
Step 1 Find the common ratio.
a5 = a4 r(5 – 4)
a5 = a4 r
–40 = –8r
5 = r
Use the given terms.
Simplify.
Substitute –40 for a5 and –8 for a4.
Divide both sides by –8.

22. Check It Out! Example 3a Continued
Step 2 Find a1.
an = a1r n - 1
–8 = a1(5)4 - 1
–0.064 = a1
General rule
Use a5 = –8 and r = 5.

23. Check It Out! Example 3a Continued
Step 3 Write the rule and evaluate for a7.
an = a1r n - 1
Substitute for a1 and r.
The 7th term is –1,000.
an = –0.064(5)n - 1
a7 = –0.064(5)7 - 1
a7 = –1,000
Evaluate for n = 7.

24. a2 = 768 and a4 = 48
Check It Out! Example 3b
Find the 7th term of the geometric sequence
with the given terms.
Step 1 Find the common ratio.
a4 = a2 r(4 – 2)
a4 = a2 r2
48 = 768r2
0.0625 = r2
Use the given terms.
Simplify.
Substitute 48 for a4 and 768 for a2.
Divide both sides by 768.
±0.25 = r Take the square root.

25. Check It Out! Example 3b Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r n - 1
768 = a1(0.25)2 - 1
3072 = a1
an = a1r n - 1
768 = a1(–0.25)2 - 1
–3072 = a1
General rule
Use a2= 768 and
r = ±0.25.
or

26. Check It Out! Example 3b Continued
Step 3 Write the rule and evaluate for a7.
Consider both the positive and negative values for r.
an = a1r n - 1
an = a1r n - 1
Substitute for
a1 and r.
an = 3072(0.25)n - 1
a7 = 3072(0.25)7 - 1
a7 = 0.75
an = 3072(–0.25)n - 1
a7 = 3072(–0.25)7 - 1
a7 = 0.75
Evaluate for
n = 7.
or

27. Check It Out! Example 3b Continued
an = a1r n - 1
an = a1r n - 1
Substitute for
a1 and r.
The 7th term is 0.75 or –0.75.
an = –3072(0.25)n - 1
a7 = –3072(0.25)7 - 1
a7 = –0.75
an = –3072(–0.25)n - 1
a7 = –3072(–0.25)7 - 1
a7 = –0.75
Evaluate for
n = 7.
or

28. Geometric means are the terms between any two
nonconsecutive terms of a geometric sequence.

29. Check It Out! Example 4
Find the geometric mean of 16 and 25.
Use the formula.

30. The indicated sum of the terms of a geometric
sequence is called a geometric series. You can
derive a formula for the partial sum of a geometric
series by subtracting the product of Sn and r from Sn
as shown.

32. Check It Out! Example 5a
Find the indicated sum for each geometric series.
Step 1 Find the common ratio.
S6 for

33. Check It Out! Example 5a Continued
Step 2 Find S6 with a1 = 2, r = , and n = 6.
Substitute.
Sum formula

34. Check It Out! Example 5b
Find the indicated sum for each geometric series.
Step 1 Find the first term.

35. Step 2 Find S6.
Check It Out! Example 5b Continued

36. Check It Out! Example 6
A 6-year lease states that the annual rent for
an office space is $84,000 the first year and
will increase by 8% each additional year of
the lease. What will the total rent expense be
for the 6-year lease?
≈ $616,218.04

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