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- 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
- 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.
- 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>
- 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
- 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>
- 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>
- 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>
- 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 .
- 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>
- 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
- 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>
- 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
- 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 .
- 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>

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