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Lesson 8-6 Geometric Sequences, page 424 Objectives: To form geometric sequences and to use formulas when describing geometric sequences. Why should we learn this? One real-world connection is to find the height of a ball after a number of bounces. (example 5) REVIEW from Chapter 5 Sequence – a number pattern Term – each number in a sequence Arithmetic sequence – add a fixed number to each term to find next term Common difference – fixed number being added to each term Definitions Geometric sequence – multiply a term by a fixed number to find next term Common ratio – fixed number being multiplied in a geometric sequence Find the common ratio of each sequence. a. 3, –15, 75, –375, . . . 3 –15 75 –375 (–5) (–5) (–5) The common ratio is –5. 3 3 3 b. 3, 2 , 4 , 8 , ... 3 3 3 3 2 4 8 1 1 1 2 2 2 1 The common ratio is 2 . 8-6 Example 1, page 424 Find the common ratio. On your own… a) 750, 150, 30, 6, … C) 4, 6, 9, 13.5, … THINK, PAIR, SHARE… b) -3, -6, -12, -24, … HINT When trying to find the common ratio, divide a term by the previous term. Repeat to find the common ratio. Example: -2, 8, -32, 128, … 8 / -2 = -4 -32 / 8 = -4 64 / -32 = -4 Common ratio = -4 Find the next three terms of the sequence 5, –10, 20, –40, . . . 5 –10 20 –40 (–2) (–2) (–2) The common ratio is –2. The next three terms are –40(–2) = 80, 80(–2) = –160, and –160(–2) = 320. 8-6 Example 2, page 425 Find the next two C) 1.1, 2.2, 4.4, 8.8 terms in the sequence. A) 1, 3, 9, 27,… B) 120, -60, 30, - 15,… Geometric or Arithmetic? Is each term being multiplied by fixed number? If so, what kind of sequence is it? Does each term have a fixed number being added? If so, what kind of sequence is it? If not, it’s neither geometric or arithmetic. Determine whether each sequence is arithmetic or geometric. a. 162, 54, 18, 6, . . . 62 54 18 6 1 1 1 3 3 3 The sequence has a common ratio. The sequence is geometric. 8-6 (continued) b. 98, 101, 104, 107, . . . 98 101 104 107 +3 +3 +3 The sequence has a common difference. The sequence is arithmetic. 8-6 Example 3, page 425 Arithmetic, Geometric or Neither? a) 2, 4, 6, 8,… c) 1, 3, 5, 7,… b) 2, 4, 8, 16, … USING A FORMULA A(n) = a • r n – 1 A(n) = nth term a = first term r = common ratio n = term number Find the first, fifth, and tenth terms of the sequence that has the rule A(n) = –3(2)n – 1. first term: A(1) = –3(2)1 – 1 = –3(2)0 = –3(1) = –3 fifth term: A(5) = –3(2)5 – 1 = –3(2)4 = –3(16) = –48 tenth term: A(10) = –3(2)10 – 1 = –3(2)9 = –3(512) = –1536 8-6 Example 4, page 426 Find the first, sixth, and twelfth terms of each sequence. A) A(n) = 4 • 3n-1 B) A(n) = -2 • 5n-1 Suppose you drop a tennis ball from a height of 2 meters. On each bounce, the ball reaches a height that is 75% of its previous height. Write a rule for the height the ball reaches on each bounce. In centimeters, what height will the ball reach on its third bounce? The first term is 2 meters, which is 200 cm. Draw a diagram to help understand the problem. 8-6 (continued) The ball drops from an initial height, for which there is no bounce. The initial height is 200 cm, when n = 1. The third bounce is n = 4. The common ratio is 75%, or 0.75. A rule for the sequence is A(n) = 200 • 0.75n – 1. Use the sequence to find the height of A(n) = 200 • 0.75n – 1 the third bounce. Substitute 4 for n to find the height of A(4) = 200 • 0.754 – 1 the third bounce. = 200 • 0.753 Simplify exponents. = 200 • 0.421875 Evaluate powers. = 84.375 Simplify. The height of the third bounce is 84.375 cm. 8-6 REAL-WORLD CONNECTION Example 5, page 426 You drop a basketball from a height of 2 meters. Each curved path has 56% of the height of the previous path. Using the height in centimeters, write a rule for the sequence. What height will the basketball reach at the top of the fourth path (where n = 4)? Round to the nearest tenth of a centimeter. Summary What did you learn today? SUMMARY Look for a pattern. Find the common difference (arithmetic) or ratio (geometric). Substitute for n to find a specific term when given a formula for a sequence. ASSIGNMENT #8-6, page 427 1-43 odd 51-71 odd

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