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 DONE BY,
‘

TN MBHAMALI
201201797
05/03/2014

TN MBHAMALI

201201797

1
SEQUENCES
AND SERIES
05/03/2014

TN MBHAMALI

201201797

2
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TN MBHAMALI

201201797

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TN MBHAMALI

201201797

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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

2...
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)...
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, ...
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...
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

...
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...
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 ...
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
s...
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...
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 ...
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...
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...
05/03/2014

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An infinite sequence is a function whose domain
is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
terms
T...
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

0...
The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of
the se...
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...
The sum of the first n terms of a sequence is
represented by summation notation.
upper limit of summation
n

ai

a1

a2

a...
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 + 32...
The sum of the terms of an infinite geometric
sequence is called a geometric series.
If |r| < 1, then the infinite geometr...
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 ...


Geometric Sequences and Series

A geometric sequence is a sequence in which each term after the first is
obtained by mu...


Finding the Common Ratio

In a geometric sequence, the common ratio can be found by dividing any
term by the term prece...


Geometric Sequences and Series

nth Term of a Geometric Sequence
In the geometric sequence with first term a1 and
commo...


Using the Formula for the nth Term

Example Find a5 and an for the geometric
sequence 4, –12, 36, –108 , …
Solution Her...


Modeling a Population of Fruit Flies

Example A population of fruit flies grows in such a
way that each generation is 1...


Geometric Series

A geometric series is the sum of the terms of a
geometric sequence .
In the fruit fly population mode...


Geometric Sequences and Series

Sum of the First n Terms of an Geometric
Sequence
If a geometric sequence has first ter...


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...
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 ...
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...
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...
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 o...
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

...
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 ...
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 ...
*** Insert one geometric mean between ¼ and 4***
*** denotes trick question
1

, ____, 4

4
a1

F irst te rm

1/4

an

n t...
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
...
Reference list
http://www.slideshare.net/bercando/sequence-and-series-11636098?qid=9878037e-6028-46c4-9246e3fb4a648965&v=q...
05/03/2014

TN MBHAMALI

201201797

47
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SEQUENCES AND SERIES

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SEQUENCES AND SERIES

  1. 1.  DONE BY, ‘ TN MBHAMALI 201201797 05/03/2014 TN MBHAMALI 201201797 1
  2. 2. SEQUENCES AND SERIES 05/03/2014 TN MBHAMALI 201201797 2
  3. 3. 05/03/2014 TN MBHAMALI 201201797 3
  4. 4. 05/03/2014 TN MBHAMALI 201201797 4
  5. 5. 05/03/2014 TN MBHAMALI 201201797 5
  6. 6. 05/03/2014 TN MBHAMALI 201201797 6
  7. 7. 05/03/2014 TN MBHAMALI 201201797 7
  8. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 20. 05/03/2014 TN MBHAMALI 201201797 20
  21. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 47. 05/03/2014 TN MBHAMALI 201201797 47

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