1. Geometric Sequences and Series
12-4
Geometric Sequences and Series
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

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

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

4. Vocabulary
geometric sequence
geometric mean
geometric series

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. Example 1A: Identifying Geometric Sequences
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
100, 93, 86, 79, ...
100, 93, 86, 79
Differences –7 –7 –7
Ratios
It could be arithmetic, with d = –7.

9. Example 1B: Identifying Geometric Sequences
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
180, 90, 60, 15, ...
180, , ,
Differences – – –45 3
Ratios 2 4
It is neither.

10. Example 1C: Identifying Geometric Sequences
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
5, 1, 0.2, 0.04, ...
5, 1, ,
Differences –4 – –0.16 5
Ratios
It could be geometric, with

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

12. 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
Ratio
It could be arithmetic, with r = –0.4.

13. 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
Ratios
It is neither.

14. 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–1r
nth term
Common
ratio
First term

15. 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.

17. Example 2: Finding the nth Term Given a Geometric Sequence
Find the 7th term of the geometric sequence 3, 12, 48, 192, ....
Step 1 Find the common ratio.
r = a2 a1 12 3
= 4 =

18. Example 2 Continued
Step 2 Write a rule, and evaluate for n = 7.
an = a1 r n–1
General rule
Substitute 3 for a1,7 for n, and 4 for r.
a7 = 3(4)7–1
= 3(4096) = 12,288
The 7th term is 12,288.

19. Check Extend the sequence.
a4 = 192
Given
a5 = 192(4) = 768
a6 = 768(4) = 3072
a7 = 3072(4) = 12,288 

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

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

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

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

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

25. Check It Out! Example 2b Continued
Check Extend the sequence.
a5 = 10
Given
a6 = 10(10) = 100
a7 = 100(10) = 1,000
a8 = 1,000(10) = 10,000
a9 = 10,000(10) = 100,000

26. Example 3: Finding the nth Term Given Two Terms
Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324.
Step 1 Find the common ratio.
a5 = a3 r(5 – 3)
Use the given terms.
a5 = a3 r2
Simplify.
Substitute 324 for a5 and 36 for a3.
324 = 36r2
9 = r2
Divide both sides by 36.
3 = r
Take the square root of both sides.

27. Example 3 Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r n - 1
an = a1r n - 1
General rule
36 = a1(3)3 - 1 or
36 = a1(–3)3 - 1
Use a3 = 36 and r = 3.
4 = a1
4 = a1

28. Example 3 Continued
Step 3 Write the rule and evaluate for a8.
Consider both the positive and negative values for r.
an = a1r n - 1
an = a1r n - 1
General rule
an = 4(3)n - 1 or
an = 4(–3)n - 1
Substitute a1 and r.
a8 = 4(3)8 - 1
a8 = 4(–3)8 - 1
Evaluate for n = 8.
a8 = 8748
a8 = –8748
The 8th term is 8748 or –8747.

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

30. 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)
Use the given terms.
a5 = a4 r
Simplify.
–40 = –8r
Substitute –40 for a5 and –8 for a4.
5 = r
Divide both sides by –8.

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

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

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

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

35. 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 or
an = 3072(–0.25)n - 1
a7 = 3072(0.25)7 - 1
a7 = 3072(–0.25)7 - 1
Evaluate for n = 7.
a7 = 0.75
a7 = 0.75

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

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

38. Example 4: Finding Geometric Means
Find the geometric mean of and .
Use the formula.

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

40. 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.

42. Example 5A: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
S8 for
Step 1 Find the common ratio.

43. Example 5A Continued
Step 2 Find S8 with a1 = 1, r = 2, and n = 8.
Sum
formula
Check Use a graphing calculator.
Substitute.

44. Example 5B: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
Step 1 Find the first term.

45. Example 5B Continued
Step 2 Find S6.
Check Use a graphing calculator.
Sum
formula
Substitute.
= 1( ) ≈ 1.97

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

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

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

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

50. Example 6: Sports Application
An online video game tournament begins with 1024 players. Four players play in each game, and in each game, only the winner advances to the next round. How many games must be played to determine the winner?
Step 1 Write a sequence.
Let n = the number of rounds,
an = the number of games played in the nth round, and
Sn = the total number of games played through n rounds.

51. Example 6 Continued
Step 2 Find the number of rounds required.
The final round will have 1 game, so substitute 1 for an.
Isolate the exponential expression by dividing by 256.
4 = n – 1
Equate the exponents.
5 = n
Solve for n.

52. Example 6 Continued
Step 3 Find the total number of games after 5 rounds.
Sum function for geometric series
341 games must be played to determine the winner.

53. 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

54. Lesson Quiz: Part I
1. Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
geometric; r = 6
2. Find the 8th term of the geometric sequence
1, –2, 4, –8, ….
–128
3. Find the 9th term of the geometric sequence with a2 = 0.3 and a6 =

55. Lesson Quiz: Part II
4. Find the geometric mean of and 18. 3
5. Find the indicated sum for the geometric series 40
6. A math tournament begins with 81 students. Students compete in groups of 3, with 1 person from each trio going on to the next round until there is 1 winner. How many matches must be played in order to complete the tournament? 40