3. DEFINITION OF SEQUENCE Copyright © by Houghton Mifflin Company, Inc. All rights reserved. An infinite sequence is a function whose domain is the set of positive integers. a 1 , a 2 , a 3 , a 4 , . . . , a n , . . . The first three terms of the sequence a n = 2 n 2 are a 1 = 2( 1 ) 2 = 2 a 2 = 2( 2 ) 2 = 8 a 3 = 2( 3 ) 2 = 18. finite sequence terms

4. DEFINITION OF GEOMETRIC SEQUENCE Copyright © by Houghton Mifflin Company, Inc. All rights reserved. A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512, . . . geometric sequence The common ratio , r , is 4 .

5. THE NTH TERM OF A GEOMETRIC SEQUENCE Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The n th term of a geometric sequence has the form a n = a 1 r n - 1 where r is the common ratio of consecutive terms of the sequence. 15, 75, 375, 1875, . . . a 1 = 1 5 The n th term is 15(5 n -1 ). a 2 = 1 5(5) a 3 = 1 5(5 2 ) a 4 = 1 5(5 3 )

6. EXAMPLE: FINDING THE NTH TERM Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example : Find the 9th term of the geometric sequence 7, 21, 63, . . . a 1 = 7 The 9th term is 45,927. a n = a 1 r n – 1 = 7(3) n – 1 a 9 = 7(3) 9 – 1 = 7(3) 8 = 7(6561) = 45,927

7. DEFINITION OF SUMMATION NOTATION Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The sum of the first n terms of a sequence is represented by summation notation . index of summation upper limit of summation lower limit of summation

8. THE SUM OF A FINITE GEOMETRIC SEQUENCE Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a 1 = 5

9. DEFINITION OF GEOMETRIC SERIES Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The sum of the terms of an infinite geometric sequence is called a geometric series . a 1 + a 1 r + a 1 r 2 + a 1 r 3 + . . . + a 1 r n -1 + . . . If | r | < 1, then the infinite geometric series has the sum

10. EXAMPLE: SUM OF INFINITE GEOMETRIC SERIES Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example : Find the sum of The sum of the series is

13. 11.3 GEOMETRIC SEQUENCES AND SERIES n th Term of a Geometric Sequence In the geometric sequence with first term a 1 and common ratio r , the n th term a n , is

17. 11.3 GEOMETRIC SEQUENCES AND SERIES Sum of the First n Terms of an Geometric Sequence If a geometric sequence has first term a 1 and common ratio r , then the sum of the first n terms is given by where .

19. Vocabulary of Sequences (Universal)

20. Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic

21. 1/2 x 9 NA 2/3

22. Find two geometric means between –2 and 54 -2, ____, ____, 54 -2 54 4 NA x The two geometric means are 6 and -18, since –2, 6, -18 , 54 forms an geometric sequence

23. -3, ____, ____, ____

24. x 9 NA

25. x 5 NA

26. *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1/4 3 NA

27. 1/2 7 x

29. 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 r < -1 No Sum Infinite Geometric -1 < r < 1

30. Find the sum, if possible:

31. Find the sum, if possible:

32. Find the sum, if possible:

33. Find the sum, if possible:

34. Find the sum, if possible:

35. The Bouncing Ball Problem – Version A A ball is dropped from a height of 50 feet. It rebounds 4/5 of it’s height, and continues this pattern until it stops. How far does the ball travel? 50 40 32 32/5 40 32 32/5

36. The Bouncing Ball Problem – Version B A ball is thrown 100 feet into the air. It rebounds 3/4 of it’s height, and continues this pattern until it stops. How far does the ball travel? 100 75 225/4 100 75 225/4

39. 11.3 INFINITE GEOMETRIC SERIES Sum of the Terms of an Infinite Geometric Sequence The sum of the terms of an infinite geometric sequence with first term a 1 and common ratio r , where –1 < r < 1 is given by .