Find the next term in each sequence. 1. 3, 6, 9, 12, … ANSWER 15 2. 0.25, 0.50, 1, 2, … ANSWER 4

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

EXAMPLE 1 Identify geometric sequences Tell whether the sequence is geometric. a. 4, 10, 18, 28, 40, . . . b. 625, 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.

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

GUIDED PRACTICE for Example 1 Tell whether the sequence is geometric. Explain why or why not. 1. 81, 27, 9, 3, 1, . . . Each ratio is , so the sequence is geometric. 1 3 ANSWER 2. 1, 2, 6, 24, 120, . . . ANSWER The ratios are different. The sequence is not geometric.

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

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. 152, – 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.

( ) ( ) 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 =

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.

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.

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 = 3072. 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

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.

( ) GUIDED PRACTICE for Examples 2, 3 and 4 Write a rule for the nth term of the geometric sequence. Then find a8. 4. 3, 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 = ; – 0.1875 1 2 n-1

( ) ( ) 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.

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.

( ) ( ) 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 = 1 + 0.059 = 1.059 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.

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 6( – 2)i–1. – 510 ANSWER

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

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

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