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### Gg

1. 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
2. 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 .
3. 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 )
4. 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
5. 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
6. 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
7. 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
8. 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
9. 11. 11.3 GEOMETRIC SEQUENCES AND SERIES <ul><li>1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. </li></ul><ul><li>The multiplier from each term to the next is called the common ratio and is usually denoted by r . </li></ul>A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.
10. 12. 11.3 FINDING THE COMMON RATIO <ul><li>In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it. </li></ul><ul><li>The geometric sequence 2, 8, 32, 128, … </li></ul><ul><li>has common ratio r = 4 since </li></ul>
11. 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
12. 14. 11.3 USING THE FORMULA FOR THE N TH TERM <ul><li>Example Find a 5 and a n for the geometric </li></ul><ul><li>sequence 4, –12, 36, –108 , … </li></ul><ul><li>Solution Here a 1 = 4 and r = 36/ –12 = – 3. Using </li></ul><ul><li>n =5 in the formula </li></ul><ul><li>In general </li></ul>
13. 15. 11.3 MODELING A POPULATION OF FRUIT FLIES <ul><li>Example A population of fruit flies grows in such a </li></ul><ul><li>way that each generation is 1.5 times the previous </li></ul><ul><li>generation. There were 100 insects in the first </li></ul><ul><li>generation. How many are in the fourth generation. </li></ul><ul><li>Solution The populations form a geometric sequence </li></ul><ul><li>with a 1 = 100 and r = 1.5 . Using n =4 in the formula </li></ul><ul><li>for a n gives </li></ul><ul><li>or about 338 insects in the fourth generation. </li></ul>
14. 16. 11.3 GEOMETRIC SERIES <ul><li>A geometric series is the sum of the terms of a geometric sequence . </li></ul><ul><li>In the fruit fly population model with a 1 = 100 and r = 1.5, the total population after four generations is a geometric series: </li></ul>
15. 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 .
16. 18. 11.3 FINDING THE SUM OF THE FIRST N TERMS <ul><li>Example Find </li></ul><ul><li>Solution This is the sum of the first six terms of a </li></ul><ul><li>geometric series with and r = 3. </li></ul><ul><li>From the formula for S n , </li></ul><ul><li> . </li></ul>
17. 19. Vocabulary of Sequences (Universal)
18. 20. Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic
19. 21. 1/2 x 9 NA 2/3
20. 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
21. 23. -3, ____, ____, ____
22. 24. x 9 NA
23. 25. x 5 NA
24. 26. *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1/4 3 NA
25. 27. 1/2 7 x
26. 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
27. 30. Find the sum, if possible:
28. 31. Find the sum, if possible:
29. 32. Find the sum, if possible:
30. 33. Find the sum, if possible:
31. 34. Find the sum, if possible:
32. 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
33. 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
34. 37. 11.3 INFINITE GEOMETRIC SERIES <ul><li>If a 1 , a 2 , a 3 , … is a geometric sequence and the </li></ul><ul><li>sequence of sums S 1 , S 2 , S 3 , …is a convergent </li></ul><ul><li>sequence, converging to a number S  . Then S  is </li></ul><ul><li>said to be the sum of the infinite geometric series </li></ul>
35. 38. 11.3 AN INFINITE GEOMETRIC SERIES <ul><li>Given the infinite geometric sequence </li></ul><ul><li>the sequence of sums is S 1 = 2, S 2 = 3, S 3 = 3.5, … </li></ul>The calculator screen shows more sums, approaching a value of 4. So
36. 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 .
37. 40. 11.3 FINDING SUMS OF THE TERMS OF INFINITE GEOMETRIC SEQUENCES <ul><li>Example Find </li></ul><ul><li>Solution Here and so </li></ul><ul><li> . </li></ul>