WarmUp Exercises Find the next term in each

Warm-Up Exercises Find the next term in each sequence. 1. 3, 6, 9, 12, … ANSWER 15 2. 0. 25, 0. 50, 1, 2, … ANSWER 4

Warm-Up Exercises 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

Warm-Up 1 Exercises EXAMPLE Identify geometric sequences Tell whether the sequence is geometric. a. 4, 10, 18, 28, 40, . . . b. 625, 125, 5, 1, . . . SOLUTION To decide whether a sequence is geometric, find the ratios of consecutive terms. a. a 2 a 4 a 5 28 10 5 a 3 = 18 = 9 14 40 = 10 = = = 10 a 1 = 4 = 2 a 3 a 4 18 28 5 9 7 ANSWER The ratios are different, so the sequence is not geometric.

Warm-Up 1 Exercises EXAMPLE Identify geometric sequences b. a 2 125 1 = = a 1 625 5 a 3 25 = 1 = a 2 125 5 a 4 5 = 1 = a 3 25 5 ANSWER Each ratio is 1 , so the sequence is geometric. 5 a 5 1 = a 4 5

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

Warm-Up Exercises 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 – 2. So the sequence is geometric.

Warm-Up 2 Exercises EXAMPLE Write a rule for the nth term of the sequence. Then find a 7. a. 4, 20, 100, 500, . . . b. 152, – 76, 38, – 19, . . . SOLUTION a. The sequence is geometric with first term a 1 = 4 and common ratio = 5. So, a rule for the nth term is: r = 20 4 an = a 1 r n – 1 = 4(5)n – 1 Write general rule. Substitute 4 for a 1 and 5 for r. The 7 th term is a 7 = 4(5)7 – 1 = 62, 500.

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

Warm-Up 3 Exercises EXAMPLE Write a rule given a term and common ratio One term of a geometric sequence is a 4 =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 = a 1 r n – 1 Write general rule. a 4 = a 1 r 4 – 1 Substitute 4 for n. 12 = a 1(2)3 Substitute 12 for a 4 and 2 for r. 1. 5 = a 1 Solve for a 1.

Warm-Up 3 Exercises EXAMPLE Write a rule given a term and common ratio So, a rule for the nth term is: an = a 1 r n – 1 Write general rule. = 1. 5(2) n – 1 Substitute 1. 5 for a 1 and 2 for r. b. 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.

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

Warm-Up 4 Exercises EXAMPLE Write a rule given two terms STEP 2 Solve the system. – 48 = a Solve Equation 1 for a 1. 2 1 r (r 5 ) Substitute for a 1 in Equation 2. 3072 = – 48 2 r 3072 = – 48 r 3 – 4 = r – 48 = a 1(– 4)2 – 3 = a 1 STEP 3 an = a 1 r n – 1 Simplify. Solve for r. Substitute for r in Equation 1. Solve for a 1. Write general rule. an = – 3(– 4)n – 1 Substitute for a 1 and r.

Warm-Up Exercises GUIDED PRACTICE for Examples 2, 3 and 4 Write a rule for the nth term of the geometric sequence. Then find a 8. 4. 3, 15, 75, 375, . . . ANSWER an = 3( 5 )n – 1; 234, 375 5. a 6 = – 96, r = 2 ANSWER an = – 3 (2)n – 1; – 384 6. a 2 = – 12, a 4 = – 3 ANSWER () 1 an = 2 n-1 ; – 0. 1875

Warm-Up 5 Exercises EXAMPLE Find the sum of a geometric series Find the sum of the geometric series 16 4(3)i – 1. i=1 a 1 = 4(3)1– 1 = 4 r=3 S 16 = a 1 1– r 16 1–r ( ) 16 1– 3 =4 1– 3 = 86, 093, 440 Identify first term. Identify common ratio. Write rule for S 16. Substitute 4 for a 1 and 3 for r. Simplify. ANSWER The sum of the series is 86, 093, 440.

Warm-Up 6 Exercises EXAMPLE 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. a. Write a rule for the total box office revenue an (in billions of dollars) in terms of the year. Let n = 1 represent 1990. b. What was the total box office revenue at U. S. movie theaters for the entire period 1990– 2003?

Warm-Up 6 Exercises EXAMPLE Use a geometric sequence and series in real life SOLUTION a. 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 a 1 = 5. 02 and r = 1 + 0. 059 = 1. 059 to write a rule for the sequence. an = 5. 02(1. 059)n – 1 Write a rule for an. b. There are 14 years in the period 1990– 2003, so find S 14. 1– (1. 059)14 1 – r 14 105 S 14 = a 1 1 – r = 5. 02 1 – 1. 059 ( ) ANSWER The total movie box office revenue for the period 1990– 2003 was about $105 billion.

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

Warm-Up Exercises 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 Warm-Up Exercises 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 a 6. ANSWER Yes; an = 5(2)n – 1; a 6 = 160 2. One term of a geometric sequence is a 5 = 48. The common ratio is r = 2. Write a rule for the nth term. ANSWER an = 3(2)n – 1

Daily Homework Quiz Warm-Up Exercises 3. Two terms of a geometric sequence are a 2 = – 2000 and a 5 = 16, 000. Find a rule for the nth term. ANSWER an = 100 (– 20)n – 1 5 4. Find the sum of the geometric series ∑ 5(3)i – 1. i=1 ANSWER 605