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TN MBHAMALI201201797

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SEQUENCES AND SERIES

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A Sequence is a set of things (usually numbers) that are in order.

a Series is a number of events, objects, or people of a similar or related kind coming one after another.

Object- to use and write sequences. To learn summutation

sequences notation

Where represents the term of the sequence

For example: write the first six terms of this sequence

hence:

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Series

When the terms in a sequences are added, the resulting expression is a series

Summation natation can be used to write a siries

The series can be written as

6 is the upper limit

greek letter sigma

1 is the lower limit

index

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Arithmetic Sequences and Series

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An introduction…………

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

Geometric Sequences

MULTIPLYTo get next term

Arithmetic Series

Sum of Terms

35

12

27.2

3 9

Geometric Series

Sum of Terms

62

20 / 3

85 / 64

9.75

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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

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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

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Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

n 1

n 1 n

nth term of arithmetic sequence

sum of n terms of arithmetic sequen

a a n 1 d

nS a a

2ce

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Given an arithmetic sequence with 15 1a 38 and d 3, find a .

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

x

15

38

NA

-3

n 1a a n 1 d

38 x 1 15 3

X = 80

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63Find S of 19, 13, 7,...

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-19

63

??

x

6

n 1a a n 1 d

?? 19 6 1

?? 353

3 6

353

n 1 n

nS a a

2

63

633 3S

219 5

63 1 1S 052

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16 1Find a if a 1.5 and d 0.5 Try this one:

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1.5

16

x

NA

0.5

n 1a a n 1 d

16 1.5 0.a 16 51

16a 9

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n 1Find n if a 633, a 9, and d 24

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

9

x

633

NA

24

n 1a a n 1 d

633 9 21x 4

633 9 2 244x X = 27

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1 29Find d if a 6 and a 20

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-6

29

20

NA

x

n 1a a n 1 d

120 6 29 x

26 28x

13x

14

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Find two arithmetic means between –4 and 5

-4, ____, ____, 5

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-4

4

5

NA

x

n 1a a n 1 d

15 4 4 x x 3

The two arithmetic means are –1 and 2, since –4, -1, 2, 5

forms an arithmetic sequence

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Find three arithmetic means between 1 and 4

1, ____, ____, ____, 4

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1

5

4

NA

x

n 1a a n 1 d

4 1 x15 3

x4

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

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Find n for the series in which 1 na 5, d 3, S 440

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

5

x

y

440

3

n 1a a n 1 d

n 1 n

nS a a

2

y 5 31x

x40 y4

25

12

x440 5 5 x 3

x 7 x440

2

3

880 x 7 3x 20 3x 7x 880

X = 16

Graph on positive window

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Geometric Sequence and Series

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An infinite sequence is a function whose domain is the set of positive integers.

a1, a2, a3, a4, . . . , an, . . .

The first three terms of the sequence an = 2n2 are

a1 = 2(1)2 = 2

a2 = 2(2)2 = 8

a3 = 2(3)2 = 18.

finite sequence

terms

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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.

82

4

328

4

12832

4

512128

4

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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.

15, 75, 375, 1875, . . . a1 = 15

The nth term is 15(5n-1).

75 515

r

a2 = 15(5)

a3 = 15(52)

a4 = 15(53)

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Example: Find the 9th term of the geometric sequence

7, 21, 63, . . .

a1 = 7

The 9th term is 45,927.

21 37

r

an = a1rn – 1 = 7(3)n – 1

a9 = 7(3)9 – 1 = 7(3)8

= 7(6561) = 45,927

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The sum of the first n terms of a sequence is represented by summation notation.

1 2 3 41

n

i ni

a a a a a a

index of summation

upper limit of summation

lower limit of summation

5

1

4n

n

1 2 3 4 54 4 4 4 4 4 16 64 256 1024 1364

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The sum of a finite geometric sequence is given by

11 1

1

1 .1

n nin

i

rS a r ar

5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?

n = 8

a1 = 5

1

81 11

221

5n

nrS ar

5210r

1 25651 2 2555

1 1275

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The sum of the terms of an infinite geometric sequence is called a geometric series.

a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .

If |r| < 1, then the infinite geometric series

11

0

.1

i

i

aS a r

r

has the sum

If 1 , then the series does not have a sum.r

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Example: Find the sum of

1

1a

Sr

1 13 13 9

13

r

3

1 13

3 31 413 3

The sum of the series is 9 .4

3 934 4

Geometric Sequences and Series

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.

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.

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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 32 128... 4

2 8 32

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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

11

nna a r

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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=5 in the formula

In general

5 1 45 4 ( 3) 4 ( 3) 324a

1 11 4 ( 3)n n

na a r

11

nna a r

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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

or about 338 insects in the fourth generation.

3 34 1 100(1.5) 337.5a a r

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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:

1 2 3 4

2 3100 100(1.5) 100(1.5) 100(1.5)

813

a a a a

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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

where .

1(1 )

1

n

n

a rS

r

1r

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Finding the Sum of the First n Terms

Example Find

Solution This is the sum of the first six terms of a

geometric series with and r = 3.

From the formula for Sn ,

.

11 2 3 6a

6

1

2 3i

i

6

6

6(1 3 ) 6(1 729) 6( 728)2184

1 3 2 2S

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Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

n 1

n 1

n1

n

nth term of geometric sequence

sum of n terms of geometric sequ

a a r

a r 1S

r 1ence

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Find the next three terms of 2, 3, 9/2, ___, ___, ___

3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3

1.5 geometric r2 3 2

3 3 3 3 3 3

2 2 2

92, 3, , , ,

2

9 9 9

2 2 2 2 2 2

92, 3, , ,

27 81 243

4 8,

2 16

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1 9

1 2If a , r , find a .

2 3

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

x

9

NA

2/3

n 1n 1a a r

9 11 2

x2 3

8

8

2x

2 3

7

8

2

3 128

6561

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Find two geometric means between –2 and 54

-2, ____, ____, 54

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

-2

54

4

NA

x

n 1n 1a a r

1454 2 x

327 x 3 x

The two geometric means are 6 and -18, since –2, 6, -18, 54

forms an geometric sequence

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2 4 1

2Find a a if a 3 and r

3

-3, ____, ____, ____

2Since r ...

3

4 83, 2, ,

3 9

2 4

8 10a a 2

9 9

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9Find a of 2, 2, 2 2,...

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

9

NA

2

2 2 2r 2

22

n 1n 1a a r

9 1

x 2 2

8

x 2 2

x 16 2

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5 2If a 32 2 and r 2, find a

____, , ____,________ ,32 2

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

5

NA

32 2

2n 1

n 1a a r 5 1

32 2 x 2

4

32 2 x 2

32 2 x4

8 2 x

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*** Insert one geometric mean between ¼ and 4***

*** denotes trick question

1,____,4

4

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/4

3

NA

4

xn 1

n 1a a r

3 114

4r 2r

14

4 216 r 4 r

1,1, 4

4

1, 1, 4

4

TN MBHAMALI 201201797 4405/03/2014

7

1 1 1Find S of ...

2 4 8

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

7

x

NA

11184r

1 1 22 4

n1

n

a r 1S

r 1

71 12 2

x12

1

1

71 12 2

12

1

63

64

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Reference list

http://www.slideshare.net/bercando/sequence-and-series-11636098?qid=9878037e-6028-46c4-9246-e3fb4a648965&v=qf1&b=&from_search=11

http://www.slideshare.net/mstfdemirdag/sequences-and-series-11032997?qid=5fcd86dc-7287-4e52-972a-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-6028-46c4-9246-e3fb4a648965&v=qf1&b=&from_search=12

http://www.slideshare.net/mcatcyonline/sequence-and-series-1902957?qid=9878037e-6028-46c4-9246-e3fb4a648965&v=default&b=&from_search=15

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THANK YOU

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