1. Find the next term in each sequence.
1. 3, 6, 9, 12, …
ANSWER 15
, 0.50, 1, 2, …
ANSWER 4

2. 3. A high school graduated 250 seniors in 2004. In 2005
the number of graduating seniors increased by 4%.
How many seniors graduated in 2005?
ANSWER
260 seniors

3. EXAMPLE 1
Identify geometric sequences
Tell whether the sequence is geometric.
a. 4, 10, 18, 28, 40, . . .
b , 125, 25, 5, 1, . . .
SOLUTION
To decide whether a sequence is geometric, find the ratios of consecutive terms. 5 2 a2 a1 = 10 4 a. a3 a2 = 18 10 9 5 a4 a3 = 28 18 14 9 a5 a4 = 40 28 10 7
ANSWER
The ratios are different, so the sequence is not geometric.

4. EXAMPLE 1
Identify geometric sequences a2 a1 =
125
625 1 5 b. 1 5 a3 a2 = 25
125 a4 a3 = 5 25 1 a5 a4 = 1 5
ANSWER
Each ratio is , so the sequence is geometric. 1 5

5. GUIDED PRACTICE
for Example 1
Tell whether the sequence is geometric. Explain why or
why not.
, 27, 9, 3, 1, . . .
Each ratio is , so the sequence is
geometric. 1 3
ANSWER
, 2, 6, 24, 120, . . .
ANSWER
The ratios are different. The sequence is
not geometric.

6. GUIDED PRACTICE
for Example 1
Tell whether the sequence is geometric. Explain why or
why not.
3. – 4, 8, – 16, 32, – 64, . . .
ANSWER
Each ratio is So the sequence is
geometric.
– 2

7. Write a rule for the nth term
EXAMPLE 2
Write a rule for the nth term
Write a rule for the nth term of the sequence. Then find a7.
a. 4, 20, 100, 500, . . .
b , – 76, 38, – 19, . . .
SOLUTION
The sequence is geometric with first term a1 = 4 and common ratio a.
r = 20 4
= 5. So, a rule for the nth term is:
an = a1 r n – 1
Write general rule.
= 4(5)n – 1
Substitute 4 for a1 and 5 for r.
The 7th term is a7 = 4(5)7 – 1 = 62,500.

8. ( ) ( ) EXAMPLE 2 Write a rule for the nth term
The sequence is geometric with first term a1 = 152 and common ratio b.
r =
–76
152 =
– 1 2
.So, a rule for the nth term is:
an = a1 r n – 1
Write general rule.
= 152
( ) 1 2 –
n – 1
Substitute 152 for a1 and for r. 1 2 –
( )
7 – 1
The 7th term is a7 = 152 1 2 – 19 8 =

9. Write a rule given a term and common ratio
EXAMPLE 3
Write a rule given a term and common ratio
One term of a geometric sequence is a4 =12. The common ratio is r = 2.
a. Write a rule for the nth term.
b. Graph the sequence.
SOLUTION
a. Use the general rule to find the first term.
an = a1r n – 1
Write general rule.
a4 = a1r 4 – 1
Substitute 4 for n.
12 = a1(2)3
Substitute 12 for a4 and 2 for r.
1.5 = a1
Solve for a1.

10. Write a rule given a term and common ratio
EXAMPLE 3
Write a rule given a term and common ratio
So, a rule for the nth term is:
an = a1r n – 1
Write general rule.
= 1.5(2) n – 1
Substitute 1.5 for a1 and 2 for r.
Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on an exponential curve. This is true for any geometric sequence with r > 0. b.

11. Write a rule given two terms
EXAMPLE 4
Write a rule given two terms
Two terms of a geometric sequence are a3 = −48 and a6 = Find a rule for the nth term.
SOLUTION
Write a system of equations using an = a1r n – 1 and substituting 3 for n (Equation 1) and then 6 for n (Equation 2).
STEP 1
a3 = a1r 3 – 1
– 48 = a1 r 2
Equation 1
a6 = a1r 6 – 1
3072 = a1r 5
Equation 2

12. Write a rule given two terms
EXAMPLE 4
Write a rule given two terms
STEP 2
Solve the system.
– 48 r2
= a1
Solve Equation 1 for a1.
3072 =
– 48 r2
(r5 )
Substitute for a1 in Equation 2.
3072 = – 48r3
Simplify.
–4 = r
Solve for r.
– 48 = a1(– 4)2
Substitute for r in Equation 1.
– 3 = a1
Solve for a1.
STEP 3
an = a1r n – 1
Write general rule.
an = – 3(– 4)n – 1
Substitute for a1 and r.

13. ( ) GUIDED PRACTICE for Examples 2, 3 and 4
Write a rule for the nth term of the geometric sequence. Then find a8.
, 15, 75, 375, . . .
ANSWER
an = 3( 5 )n – 1; 234,375
5. a6 = – 96, r = 2
ANSWER
an = – 3 (2)n – 1; – 384
6. a2 = – 12, a4 = – 3 ( )
ANSWER
an = ; – 1 2
n-1

14. ( ) ( ) EXAMPLE 5 Find the sum of a geometric series
Find the sum of the geometric series 16
i = 1
4(3)i – 1.
a1 = 4(3)1– 1 = 4
Identify first term.
r = 3
Identify common ratio.
S16 = a1
1– r16
1 – r
( )
Write rule for S16.
= 4
1– 316
1 – 3
( )
Substitute 4 for a1 and 3 for r.
= 86,093,440
Simplify.
ANSWER
The sum of the series is 86,093,440.

15. EXAMPLE 6
Use a geometric sequence and series in real life
Movie Revenue
In 1990, the total box office revenue at U.S. movie theaters was about $5.02 billion. From 1990 through 2003, the total box office revenue increased by about 5.9% per year.
Write a rule for the total box office revenue an (in billions of dollars) in terms of the year. Let n = 1 represent 1990. a.
What was the total box office revenue at U.S. movie theaters for the entire period 1990–2003? b.

16. ( ) ( ) EXAMPLE 6 Use a geometric sequence and series in real life
SOLUTION
Because the total box office revenue increased by the same percent each year, the total revenues from year to year form a geometric sequence. Use a1 = 5.02 and r = = to write a rule for the sequence. a.
an = 5.02(1.059)n – 1
Write a rule for an.
There are 14 years in the period 1990–2003, so find S14. b.
= 5.02
1– (1.059)14
1 – 1.059
( )
1 – r14
1 – r
( )
S14 = a1
105
ANSWER
The total movie box office revenue for the period 1990–2003 was about $105 billion.

17. 7. Find the sum of the geometric series 6( – 2)i–1.
GUIDED PRACTICE
for Examples 5 and 6 8
i – 1
7. Find the sum of the geometric series ( – 2)i–1.
– 510
ANSWER

18. GUIDED PRACTICE
for Examples 5 and 6
8. MOVIE REVENUE Use the rule in part (a) of Example 6 to estimate the total box office revenue at U.S. movie theaters in 2000.
ANSWER
about $8.91 billion

19. Daily Homework Quiz
1. Tell whether the sequence 5, 10, 20, 40, is geometric. If so, write a rule for the nth term of the sequence and find a6.
ANSWER
Yes; an = 5(2)n – 1; a6 = 160
2. One term of a geometric sequence is a5 = 48. The common ratio is r = 2. Write a rule for the nth term.
ANSWER
an = 3(2)n – 1

20. Daily Homework Quiz
3. Two terms of a geometric sequence are a2 = –2000 and a5 = 16,000,000. Find a rule for the nth term.
ANSWER
an = 100 (–20)n – 1
4. Find the sum of the geometric series 5(3)i – 1. ∑
i = 1 5
605
ANSWER