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- 1. DONE BY, ‘ TN MBHAMALI 201201797 05/03/2014 TN MBHAMALI 201201797 1
- 2. SEQUENCES AND SERIES 05/03/2014 TN MBHAMALI 201201797 2
- 3. 05/03/2014 TN MBHAMALI 201201797 3
- 4. 05/03/2014 TN MBHAMALI 201201797 4
- 5. 05/03/2014 TN MBHAMALI 201201797 5
- 6. 05/03/2014 TN MBHAMALI 201201797 6
- 7. 05/03/2014 TN MBHAMALI 201201797 7
- 8. An introduction………… 1, 4, 7, 10, 13 9, 1, 7, 3, 6 2, 4, 8, 16, 32 2 7 .2 3 62 9, 12 15 6.2, 6.6, 7, 7.4 , 35 20 / 3 3, 1, 1/ 3 1, 1 / 4, 1 / 16, 1 / 64 8 5 / 6 4 9 , 2.5 , 6.25 Arithmetic Sequences Geometric Sequences ADD To get next term 9 .7 5 MULTIPLY To get next term Arithmetic Series Sum of Terms 05/03/2014 TN MBHAMALI Geometric Series Sum of Terms 201201797 8
- 9. Find the next four terms of –9, -2, 5, … Arithmetic Sequence 2 9 5 2 7 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33 05/03/2014 TN MBHAMALI 201201797 9
- 10. Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k 05/03/2014 TN MBHAMALI 201201797 10
- 11. Vocabulary of Sequences (Universal) a1 F irst te rm an n th te rm n Sn d num ber of term s sum of n term s com m on difference nth term of arithm etic sequence an sum of n term s of arithm etic sequen ce 05/03/2014 TN MBHAMALI a1 n Sn 1 d n 2 201201797 a1 an 11
- 12. Given an arithmetic sequence with a 15 x a1 an n th te rm 15 n NA Sn 3, find a 1. F irst te rm 38 38 and d -3 d num ber of term s sum of n term s com m on difference an a1 n 38 x 15 1 d 1 3 X = 80 05/03/2014 TN MBHAMALI 201201797 12
- 13. Find S 63 of 19, 13, 7,... -19 F irst te rm ?? an n th te rm 63 353 a1 n x an ?? ?? a1 19 n Sn 6 num ber of term s d sum of n term s com m on difference 1 d 63 Sn 1 6 S 63 353 S 63 05/03/2014 TN MBHAMALI n 2 a1 63 an 19 353 2 1 052 1 201201797 13
- 14. Try this one: Find a 16 if a 1 1.5 and d 0.5 1.5 a1 F irst te rm x an n th te rm 16 n NA Sn 0.5 d num ber of term s sum of n term s com m on difference an a 16 1.5 a 16 05/03/2014 a1 9 TN MBHAMALI n 16 1 d 1 0. 5 201201797 14
- 15. F ind n if a n 633, a 1 9, and d 24 9 a1 F irst te rm 633 an n th te rm x n NA Sn 24 d num ber of term s sum of n term s com m on difference an a1 n 1 d 633 9 x 1 24 633 9 24x 24 X = 27 05/03/2014 TN MBHAMALI 201201797 15
- 16. Find d if a 1 6 and a 29 20 -6 a1 F irst te rm 20 an n th te rm 29 n NA Sn x d an num ber of term s sum of n term s com m on difference a1 n 20 6 29 26 28 x x 1 d 1 x 13 14 05/03/2014 TN MBHAMALI 201201797 16
- 17. Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 a1 F irst te rm 5 an n th te rm 4 n NA x num ber of term s Sn sum of n term s d com m on difference an a1 n 1 d 5 4 4 1 x x 3 The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence 05/03/2014 TN MBHAMALI 201201797 17
- 18. Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 a1 F irst te rm 4 an n th te rm 5 n NA x num ber of term s Sn sum of n term s d com m on difference an a1 4 1 x n 5 1 d 1 x 3 4 The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence 05/03/2014 TN MBHAMALI 201201797 18
- 19. Find n for the series in which a 1 5 a1 F irst te rm y an n th te rm x n 5, d 440 Sn 3 d an y 440 3, S n x 5 440 x 7 x 1 3 com m on difference 1 d 3x 2 sum of n term s n 5 2 num ber of term s a1 440 880 0 x 7 3x 2 3x 7x 880 Graph on positive window 5 x 1 3 X = 16 Sn 440 n 2 x a1 5 an y 2 05/03/2014 TN MBHAMALI 201201797 19
- 20. 05/03/2014 TN MBHAMALI 201201797 20
- 21. An infinite sequence is a function whose domain is the set of positive integers. a1, a2, a3, a4, . . . , an, . . . terms The first three terms of the sequence an = 2n2 are a1 = 2(1)2 = 2 a2 = 2(2)2 = 8 finite sequence a3 = 2(3)2 = 18. 05/03/214 TN MBHAMALI 201201797 21
- 22. A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512, . . . 8 2 32 8 4 512 128 05/03/2014 4 128 32 geometric sequence 4 4 The common ratio, r, is 4. TN MBHAMALI 201201797 22
- 23. The nth term of a geometric sequence has the form an = a1rn - 1 where r is the common ratio of consecutive terms of the sequence. r a1 = 15 75 15 5 15, 75, 375, 1875, . . . a2 = 15(5) a3 = a4 = 15(52) 15(53) The nth term is 15(5n-1). 05/03/2014 TN MBHAMALI 201201797 23
- 24. Example: Find the 9th term of the geometric sequence 7, 21, 63, . . . a1 = 7 r 21 7 3 an = a1rn – 1 = 7(3)n – 1 a9 = 7(3)9 – 1 = 7(3)8 = 7(6561) = 45,927 The 9th term is 45,927. 05/03/2014 TN MBHAMALI 201201797 24
- 25. The sum of the first n terms of a sequence is represented by summation notation. upper limit of summation n ai a1 a2 a3 a4 an i 1 lower limit of summation index of summation 5 4 n 1 n 4 4 1 4 16 2 4 64 3 4 4 256 4 5 1024 1364 05/03/2014 TN MBHAMALI 201201797 25
- 26. The sum of a finite geometric sequence is given by n Sn a1r n a1 1 r . 1 r i 1 i 1 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n=8 r a1 = 5 n Sn 05/03/2014 a1 1 r 1 r 8 5 1 2 1 2 TN MBHAMALI 5 1 256 1 2 201201797 10 5 2 5 255 1 1275 26
- 27. The sum of the terms of an infinite geometric sequence is called a geometric series. If |r| < 1, then the infinite geometric series a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . . has the sum S a1r i 0 If r 05/03/2014 a1 i 1 r . 1 , then the series does not have a su m. TN MBHAMALI 201201797 27
- 28. Example: Find the sum of 3 1 1 3 a1 1 1 3 r S 1 9 3 r 1 3 1 3 1 1 3 3 4 3 3 3 4 9 4 The sum of the series is 9 . 4 05/03/2014 TN MBHAMALI 201201797 28
- 29. Geometric Sequences and Series 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. 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. The multiplier from each term to the next is called the common ratio and is usually denoted by r. 05/03/2014 TN MBHAMALI 201201797 29
- 30. Finding the Common Ratio In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it. The geometric sequence 2, 8, 32, 128, … has common ratio r = 4 since 8 128 2 05/03/2014 32 8 32 TN MBHAMALI ... 4 201201797 30
- 31. Geometric Sequences and Series nth Term of a Geometric Sequence In the geometric sequence with first term a1 and common ratio r, the nth term an, is an 05/03/2014 TN MBHAMALI a1 r n 1 201201797 31
- 32. Using the Formula for the nth Term Example Find a5 and an for the geometric sequence 4, –12, 36, –108 , … Solution Here a1= 4 and r = 36/ –12 = – 3. Using n 1 n=5 in the formula a n a1 r a5 4 ( 3) 5 1 4 ( 3) 4 324 In general an 05/03/2014 a1 r n 1 TN MBHAMALI 4 ( 3) n 1 201201797 32
- 33. Modeling a Population of Fruit Flies Example A population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation. Solution The populations form a geometric sequence with a1= 100 and r = 1.5 . Using n=4 in the formula for an gives a4 a1 r 3 100(1.5) 3 337.5 or about 338 insects in the fourth generation. 05/03/2014 TN MBHAMALI 201201797 33
- 34. Geometric Series A geometric series is the sum of the terms of a geometric sequence . In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series: a1 a2 100 a3 a4 100(1.5) 100(1.5) 2 100(1.5) 3 813 05/03/2014 TN MBHAMALI 201201797 34
- 35. Geometric Sequences and Series Sum of the First n Terms of an Geometric Sequence If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by Sn 05/03/2014 a1 (1 1 n r ) r TN MBHAMALI where r 1 201201797 . 35
- 36. Finding the Sum of the First n Terms 6 Example Find 2 3 i i 1 Solution This is the sum of the first six terms of a 1 6 and r = 3. geometric series with a1 2 3 From the formula for Sn , S6 05/03/2014 6 (1 1 6 3 ) 6 (1 3 TN MBHAMALI 729) 2 6 ( 7 2 8) 2 201201797 2184 . 36
- 37. Vocabulary of Sequences (Universal) a1 F irst te rm an n th te rm n num ber of term s Sn r sum of n term s com m on ratio nth term of geom etric sequence an sum of n term s of geom etric sequ ence 05/03/2014 TN MBHAMALI a 1r n 1 Sn 201201797 a1 r r n 1 1 37
- 38. Find the next three terms of 2, 3, 9/2, ___, ___, ___ 3 – 2 vs. 9/2 – 3… not arithmetic 3 9/2 2 3 1 .5 g e o m e tric 3 r 2 9 9 2, 3, , 2 2 3 9 , 2 2 3 2 3 9 , 2 2 3 3 3 2 2 2 9 27 81 243 2, 3, , , , 2 4 8 16 05/03/2014 TN MBHAMALI 201201797 38
- 39. If a 1 1 2 ,r 2 3 , fin d a 9 . a1 an n th te rm x num ber of term s r NA com m on ratio Sn 9 sum of n term s n 2/3 an a 1r n 1 1 2 9 1 3 2 x TN MBHAMALI 2 2 x 05/03/2014 1/2 F irst te rm 8 3 2 8 7 3 8 128 6561 201201797 39
- 40. Find two geometric means between –2 and 54 -2, ____, ____, 54 a1 an n Sn r -2 F irst te rm 54 n th te rm num ber of term s sum of n term s com m on ratio 4 an a 1r 54 NA n 1 2 x 27 3 x x 4 1 3 x The two geometric means are 6 and -18, since –2, 6, -18, 54 forms an geometric sequence 05/03/2014 TN MBHAMALI 201201797 40
- 41. F in d a 2 a 4 if a 1 3 and r 2 3 -3, ____, ____, ____ 2 S in c e r ... 3 3, 2, 4 8 , 3 a2 05/03/2014 a4 2 TN MBHAMALI 9 8 9 10 9 201201797 41
- 42. F in d a 9 o f 2 , 2, 2 2 ,... a1 F irst te rm an n th te rm x num ber of term s 9 n Sn r 2 NA sum of n term s com m on ratio r an a 1r 2 2 n 1 2 2 2 2 9 1 x 2 2 8 x x 05/03/2014 2 16 2 TN MBHAMALI 2 201201797 42
- 43. If a 5 32 2 and r 2 , find a 2 ____, ____ , ____, ____ ,32 2 a1 an x F irst te rm n th te rm n Sn r 32 2 num ber of term s NA sum of n term s 2 com m on ratio an a 1r 5 n 1 5 1 32 2 x 2 4 32 2 x 32 2 4x 8 2 05/03/2014 TN MBHAMALI 2 x 201201797 43
- 44. *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1 , ____, 4 4 a1 F irst te rm 1/4 an n th te rm 4 num ber of term s 3 n sum of n term s NA com m on ratio x Sn r an 4 1 4 05/03/2014 r 3 1 4 1 r 2 a 1r 16 r 2 1 n 1 4 4 r 1 4 TN MBHAMALI , 1, 4 , 1, 4 4 201201797 44
- 45. F in d S 7 o f 1 1 1 2 4 8 ... a1 F irst te rm 1/2 an n th te rm NA n r Sn a1 r r x 1 r com m on ratio 1 4 1 8 1 4 1 2 1 1 2 7 2 1 2 05/03/2014 sum of n term s 1 1 x 7 2 Sn n num ber of term s 1 1 2 1 7 2 1 63 1 1 64 2 TN MBHAMALI 201201797 45
- 46. Reference list http://www.slideshare.net/bercando/sequence-and-series-11636098?qid=9878037e-6028-46c4-9246e3fb4a648965&v=qf1&b=&from_search=11 http://www.slideshare.net/mstfdemirdag/sequences-and-series-11032997?qid=5fcd86dc-7287-4e52972a-c8392094a20d&v=qf1&b=&from_search=6 https://www.google.co.za/#q=what+is+a+sequence http://www.slideshare.net/jfuller2012/sequences-and-series-6125259?qid=9878037e-6028-46c49246-e3fb4a648965&v=qf1&b=&from_search=12 http://www.slideshare.net/mcatcyonline/sequence-and-series-1902957?qid=9878037e-6028-46c49246-e3fb4a648965&v=default&b=&from_search=15 TN MBHAMALI 201201797 46
- 47. 05/03/2014 TN MBHAMALI 201201797 47

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