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Geometric Sequences and Series

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Geometric Sequences and Series Slide 2 Arithmetic Sequences ADD To get next term Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms Geometric Series Sum of Terms 2. Geometric Sequences and Series a1a1 a2a2 a3a3 a4a4 a5a5 a6a6 anan a n - 1 + d r r r r r r Slide 3 Geometric Sequences (Type 2) In geometric sequences, you multiply by a common ratio (r) each time. 1, 2, 4, 8, 16,... multiply by 2 27, 9, 3, 1, 1/3,... Divide by 3 which means multiply by 1/3 ie Slide 4 The nth term of an geometric sequence is denoted by the formula Where a is the 1 st term and r is the common ratio The sum of the first n terms of a geometric series is found by using: Note if r>1 then we can use the formula Which is more convenient Slide 5 Vocabulary of Sequences (Universal) Slide 6 Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 2 vs. 9/2 3 not arithmetic Slide 7 1/2 x 9 NA 2/3 Slide 8 Example 2 Find u 10 for the geometric sequence 144, 108, 81, 60, Answer a = 144 and r = u 10 = ar n-1 = 144 () 9 = 10.812 Slide 9 Example 3 Find S 19 for the geometric sequence 3-6+12-24+ Answer a = 3 and r = Slide 10 -3, ____, ____, ____ Slide 11 x 9 NA Slide 12 x 5 Slide 13 1/2 7 x Slide 14 1, 4, 7, 10, 13, . Infinite Arithmetic No Sum 3, 7, 11, , 51 Finite Arithmetic 1, 2, 4, , 64 Finite Geometric 1, 2, 4, 8, Infinite Geometric r 1 or r -1 No Sum Infinite Geometric -1 < r < 1 13.5 Infinite Geometric Series |r| 1 |r| < 1 Slide 15 Find the sum, if possible: Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 The Bouncing Ball Problem Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of its height, and continues this pattern until it stops. How far does the ball travel? 50 40 32 32/5 40 32 32/5 Slide 21 The Bouncing Ball Problem Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of its height, and continues this pattern until it stops. How far does the ball travel? 100 75 225/4 100 75 225/4 Slide 22 An old grandfather clock is broken. When the pendulum is swung it follows a swing pattern of 25 cm, 20 cm, 16 cm, and so on until it comes to rest. What is the total distance the pendulum swings before coming to rest? 25 20 16 25 20 16