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

2. Vocabulary
geometric sequence
geometric mean
geometric series

3. 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, ...

4. 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, ...

5. 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, ...

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

7. 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, . . .

8. 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

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

10. 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.

11. Example 3 Continued
Step 2 Find a1.
Consider both the positive and negative values for r.
an = a1r n - 1
an = a1r n - 1
36 = a1(3)3 - 1 or
36 = a1(–3)3 - 1

12. 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
an = 4(3)n - 1 or
an = 4(–3)n - 1

13. 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.

14. Check It Out! Example 3a
Find the 7th term of the geometric sequence with the given terms.
a4 = –8 and a5 = –40

15. Check It Out! Example 3b
Find the 7th term of the geometric sequence with the given terms.
a2 = 768 and a4 = 48

16. 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.

18. Example 5A: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.
S8 for

19. 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.

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

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

22. Example 5B: Finding the Sum of a Geometric Series
Find the indicated sum for the geometric series.

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

24. Example 5B Continued
Step 2 Find S6.
Check Use a graphing calculator.
Sum
formula
Substitute.

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

26. Check It Out! Example 5a
Find the indicated sum for each geometric series.
S6 for

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

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

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

30. Check It Out! Example 5b
Find the indicated sum for each geometric series.

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

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

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

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

35. Lesson Quiz: Part II
4. Find the geometric mean of and 18.
5. Find the indicated sum for the geometric series
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?