Objective: Learn to recognize and find terms in a geometric sequence. =

Count start at 0. Count start at 1.

GEOMETRIC SEQUENCE RULE In a geometric sequence, the ratio of one term to the next is always the same. This ratio is called the common ratio (r). The common ratio is multiplied by each term to get the next term. GEOMETRIC SEQUENCE RULE The nth term an of a geometric sequence with common ratio r is Also written as:

In an arithmetic sequence, the terms are found by adding a constant amount to the preceding term. In a geometric sequence, the terms are found by multiplying each term after the first by a constant amount. This constant multiplier is called the common ratio and is denoted r.

Example:

Example 1: Identifying Geometric Sequences Determine if the sequence could be geometric. If so, give the common ratio. A. 1, 5, 25, 125, 625, … Divide each term by the term before it. 1 5 25 125 625, . . . 5 5 5 5 The sequence could be a geometric with a common ratio of 5.

Example 1: Whiteboard Determine if the sequence could be geometric. If so, give the common ratio. A. 2, 10, 50, 250, 1250, . . . Divide each term by the term before it. 2 10 50 250 1250, . . . 5 5 5 5 The sequence could be a geometric with a common ratio of 5.

Example 2: Whiteboard Determine if the sequence could be geometric. If so, give the common ratio. B. 1, 3, 9, 12, 15, … Divide each term by the term before it. 1 3 9 12 15, . . . 3 3 43 54 The sequence is not geometric.

Example 3: Whiteboard Determine if the sequence could be geometric. If so, give the common ratio. B. 1, 1, 1, 1, 1, . . . Divide each term by the term before it. 1 1 1 1 1, . . . 1 1 1 1 The sequence could be a geometric with a common ratio of 1.

Example 4: Whiteboard Determine if the sequence could be geometric. If so, give the common ratio. C. 81, 27, 9, 3, 1, . . . Divide each term by the term before it. 81 27 9 3 1, . . . 13 13 13 13 The sequence could be geometric with a common ratio of . 1 3

Example 5: Whiteboard Determine if the sequence could be geometric. If so, give the common ratio. C. 2, 4, 12, 24, 96, . . . Divide each term by the term before it. 2 4 12 24 96, . . . 2 3 2 4 The sequence is not geometric.

Example 1: Finding a Given Term of a Geometric Sequence Find the given term in the geometric sequence. A. 11th term: –2, 4, –8, 16, . . . r = = –2 4 –2 an = a1rn–1 a11 = –2(–2)10 = –2(1024) = –2048

Example 6: Whiteboard Find the given term in the geometric sequence. A. 12th term: -2, 4, -8, 16, . . . r = = –2 4 –2 an = a1rn–1 a12 = –2(–2)11 = –2(–2048) = 4096

Example 7: Whiteboard Find the given term in the geometric sequence. B. 9th term: 100, 70, 49, 34.3, . . . r = = 0.7 70 100 an = a1rn–1 a9 = 100(0.7)8 = 100(0.05764801) = 5.764801

Example 8: Whiteboard Find the given term in the geometric sequence. B. 11th term: 100, 70, 49, 34.3, . . . r = = 0.7 70 100 an = a1rn–1 a11 = 100(0.7)10 = 100(0.0282475249)  2.825

Example 9: Whiteboard Find the given term in the geometric sequence. C. 10th term: 0.01, 0.1, 1, 10, . . . r = = 10 0.1 0.01 an = a1rn–1 a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000

Example 10: Whiteboard Find the given term in the geometric sequence. C. 5th term: 0.01, 0.1, 1, 10, . . . r = = 10 0.1 0.01 an = a1rn–1 a5 = 0.01(10)4 = 0.01(10,000) = 100

Example 11: Whiteboard Find the given term in the geometric sequence. D. 7th term: 1000, 200, 40, 8, . . . r = = 200 1000 1 5 an = a1rn–1 a7 = 1000( )6 = 1000( )= , or 0.064 1 5 8 125 15,625

Example 12: Whiteboard Find the given term in the geometric sequence. D. 12th term: 1000, 200, 40, 8, … r = = 200 1000 1 5 an = a1rn–1 a5 = 1000 ( )4 = 1000( )= , or 1.6 1 5 8 625

Example 1: Money Application Tara sells computers. She has the option of earning (1) $50 per sale or (2) $1 for the first sale, $2 for the second sale, $4 for the third sale and so on, where each sale is worth twice as much as the previous sale. If Tara estimates that she can sell 10 computers a week, which option should she choose? If Tara chooses $50 per sale, she will get a total of 10($50) = $500.

Example 1 Continued If Tara chooses the second option, her earnings for just the 10th sale will be more that the total of all the earnings in option 1. a10 = ($1)(2)9 = ($1)(512) = $512 Option 1 gives Tara more money in the beginning, but option 2 gives her a larger total amount.

Example 1: Whiteboard A gumball machine at the mall has 932 gumballs. If 19 gumballs are bought each day, how many gumballs will be left in the machine on the 7th day? n = 7 a1 = 932 r =  0.98 913 932 an = a1rn–1 a7 = (932)(0.98)6  (932)(0.89)  829 There will be about 829 gumballs in the machine after 7 days.