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Page 1: Sequences and Series - NJCTLcontent.njctl.org/courses/math/pre-calculus/sequences... · 2015-12-04 · Arithmetic Series An arithmetic series is the sum of the terms in the arithmetic

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Page 2: Sequences and Series - NJCTLcontent.njctl.org/courses/math/pre-calculus/sequences... · 2015-12-04 · Arithmetic Series An arithmetic series is the sum of the terms in the arithmetic

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Pre­Calculus

Sequences and Series

www.njctl.org

2015­03­24

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Table of Contents

Arithmetic Sequences

Arithmetic Series

Geometric Sequences

Geometric Series

Special Sequences

Infinite Geometric Series

Binomial Theorem

click on the topic to go to that section

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Arithmetic Sequences

Return to Table of Contents

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Arithmetic Sequence ­ adding the same value to get from term to term.

Find the next three terms:

1, 4, 7, 10, . . .

5, 11, 17, 23, . . .

9, 5, 1, ­3, . . .

Arithmetic Sequence

Teacher

Teacher

If you are unsure of what is being added, you can find the common difference, d, by subtracting the first term from the second.

PullPull

for extra hint

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6

Arithmetic SequenceThe value being added or the common difference between terms is represented by the variable d.

Find d:

1, 4, 7, 10, . . .

5, 11, 17, 23, . . .

9, 5, 1, ­3, . . .

Arithmetic Sequence

Teacher

Teacher

If you are unsure of what is being added, you can find the common difference, d, by subtracting the first term from the second.

PullPull

for extra hint

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1 Find the next term in the arithmetic sequence: 3, 9, 15, 21, . . .

Arithmetic Sequence

Teacher

Teacher

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2 Find the next term in the arithmetic sequence: ­8, ­4, 0, 4, . . .

Arithmetic Sequence

Teacher

Teacher

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3 Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9, . . .

Arithmetic Sequence

Teacher

Teacher

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4 Find the value of d in the arithmetic sequence: 10, ­2, ­14, ­26, . . .

Arithmetic Sequence

Teacher

Teacher

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5 Find the value of d in the arithmetic sequence: ­8, 3, 14, 25, . . .

Arithmetic Sequence

Teacher

Teacher

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Arithmetic SequenceAs we study sequences we need a way of naming the terms. a1 to represent the first term,a2 to represent the second term,a3 to represent the third term,and so on in this manner.

If we were talking about the 8th term we would use a8.

When we want to talk about general term call it the nth termand use an.

Arithmetic Sequence

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Arithmetic SequenceWrite the first four terms of the arithmetic sequence that is described.

a1 = 4; d = 6

a1 = 3; d = ­3

a1 = 0.5; d = 2.3

a2 = 7; d = 5

Arithmetic Sequence

Teacher

Teacher

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6 Which sequence matches the description?

A 4, 6, 8, 10

B 2, 6,10, 14

C 2, 8, 32, 128

D 4, 8, 16, 32

Arithmetic Sequence

Teacher

Teacher

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7 Which sequence matches the description?

A ­3, ­7, ­10, ­14

B ­4, ­7, ­11, ­13

C ­3, ­7, ­11, ­15

D ­3, 1, 5, 9

Arithmetic Sequence

Teacher

Teacher

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8 Which sequence matches the description?

A 7, 10, 13, 16

B 4, 7, 10, 13

C 1, 4, 7,10

D 3, 5, 7, 9

Arithmetic Sequence

Teacher

Teacher

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Arithmetic Sequence

To find a specific term,say the 5th or a5, you couldwrite out all of the terms.

But what about the 100th term(or a100 )?

We need to find a formula to get there directly without writing out the whole list.

Arithmetic Sequence

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Arithmetic Sequence

Consider: 3, 9, 15, 21, 27, 33, 39,. . .

a1 3

a2

a3

a4

a5

a6

a7

Do you see a pattern that relates the term number to its

value?

Arithmetic Sequence

Teacher

Teacher

click

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Arithmetic SequenceExample Find the 21st term of the arithmetic sequence with a1 = 4 and d = 3.

Example Find the 12th term of the arithmetic sequence with a1 = 6 and d = ­5.

Arithmetic Sequence

Teacher

Teacher Solution: an = a1 +(n­1)d

a21 = 4 + (21 ­ 1)3 a21 = 4 + (20)3 a21 = 4 + 60 a21 = 64

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Arithmetic SequenceExample Find the 1st term of the arithmetic sequence with a15 = 30 and d = 7.

Example Find the 1st term of the arithmetic sequence with a17 = 4 and d = ­2.

Arithmetic Sequence

Teacher

Teacher

Solution: an = a1 +(n ­1)d 30 = a1 + (15 ­ 1)7

30 = a1 + (14)7 30 = a1 + 98 ­58 = a1

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21

Arithmetic SequenceExample Find d of the arithmetic sequence with a15 = 42 and a1=3.

Example Find the term number n of the arithmetic sequence with an = 6, a1=­34 and d = 4.

Arithmetic Sequence

Teacher

Teacher

Solution: an = a1 +(n ­1)d 45 = 3 + (15 ­ 1)d

45 = 3 + (14)d 42 = 14d 3 = d

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9 Find a11 when a1 = 13 and d = 6.

Arithmetic Sequence

Teacher

Teacher

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23

10 Find a17 when a1 = 12 and d = ­0.5

Arithmetic Sequence

Teacher

Teacher

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11 Find a17 for the sequence 2, 4.5, 7, 9.5, ...

Arithmetic Sequence

Teacher

Teacher

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12 Find the common difference d when a1 = 12 and a13= 6.

Arithmetic Sequence

Teacher

Teacher

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13 Find n such a1 = 12 , an= ­20, and d = ­2.

Arithmetic Sequence

Teacher

Teacher

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14 Tom works at a car dealership selling cars. He is paid $4000 a month plus a $300 commission for every car he sells after the first car . How much did he make in April if he sold 14 cars?

Arithmetic Sequence

Teacher

Teacher

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Arithmetic Sequence

Find the missing terms in the arithmetic sequence.

4, 6, 8, 10,___

5, 10, ___, 20

___, 12, 9, 6

6,___, 14

Notice in the last example d was added to get from 6 to ___ and another d was added to get from ___ to 14.

Or 6 + 2d = 14

Arithmetic Sequence

Teacher

Teacher

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Arithmetic Sequence

Find the missing terms

2, ___ , ___ , 23

4, ___ , ___ , ­14

7, ___ , ___, ___, 39

­9, ___ , ___ , ___, ___ , ­34

Arithmetic Sequence

Teacher

Teacher

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15 Find the missing term: 4, ___ , ­16

A ­20

B ­10

C ­6

D 2

Arithmetic Sequence

Teacher

Teacher

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16 Find the missing terms: ­10, ___ , ___, 8

A ­6, ­2

B ­6, 2

C ­5, 1

D ­4, 2

Arithmetic Sequence

Teacher

Teacher

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17 Find the missing terms: 12, ___ , ___, 75

A 27, 51

B 33, 54

C 37, 51

D 34, 58

Arithmetic Sequence

Teacher

Teacher

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18 Find d for the arithmetic sequence: 5, ___ , ___ , ___ , 21

Arithmetic Sequence

Teacher

Teacher

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34

Arithmetic Series

Return to Table of Contents

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Arithmetic SeriesAn arithmetic series is the sum of the terms in the arithmetic sequence.

Sn represents the sum of the first n terms.

Consider 4, 7,10, 13, 16, 19, 22, . . .

S4= 4 + 7 + 10 + 13 = 34

S6= 4 + 7 + 10 + 13 + 16 + 19= 69

Arithmetic Series

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19 Consider 3, 9, 15, 21, 27, 33, 39,... what is S4?

Arithmetic Series

Teacher

Teacher

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20 Consider 3, 9, 15, 21, 27, 33, 39,... what is S5?

Arithmetic Series

Teacher

Teacher

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21 Consider 3, 9, 15, 21, 27, 33, 39,... what is S7?

Arithmetic Series

Teacher

Teacher

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Arithmetic SeriesSuppose we wanted to find the first 100 terms of 4, 7,10, 13, 16, 19, 22, . . . or S 100?

There must be a short cut.

The 100th term of the sequence is 301 using a100 = 4 + (100 ­ 1)3.

S100 = 4 + 7 + 10 +13 + . . . + 292 + 295 +298 + 301If we add the smallest and largest (4 + 301)= 305If we add the next two (7 + 298) = 305and continue (10 + 295) until they are all paired up (151+154).

We now have 50 pairs of 305, so...S100 = 4 + 7 + 10 +. . .+298 + 301 = 50(305) = 15250

Do you see a pattern?

Arithmetic Series

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Arithmetic SeriesS100 = 4 + 7 + 10 +. . .+298 + 301 = 50(305) = 15250

To use the formula, you'll need the first term, the last term, and the number of terms. If one of those is missing, use

an = a1 + (n ­ 1)d to find it.

Example: Find Sn if a1 = 16, an = 104,and n = 20

Arithmetic Series

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Find Sn for each arithmetic series.

Example: a1 = 7 and a12 = ­23

Example: a1 = 6, n = 10, and d = 9

Arithmetic Series

Teacher

Teacher

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Find Sn for each arithmetic series.Example: a12= 30 and d = ­7

Example: a1 = 2, an = 32, and d = 5

Arithmetic Series

Teacher

Teacher

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43

22 Find the Sn for the arithmetic series described:a1 = 19 and a12 = 37.

Arithmetic Series

Teacher

Teacher

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23 Find the Sn for the arithmetic series described:a1 = 30 and a17 = ­45.

Arithmetic Series

Teacher

Teacher

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45

24 Find the Sn for the arithmetic series described:a1 = 20, n = 8, and d = 6.

Arithmetic Series

Teacher

Teacher

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25 Find the Sn for the arithmetic series described:an = 20, n = 9, and d = ­4.

Arithmetic Series

Teacher

Teacher

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26 Find the Sn for the arithmetic series described:17 + 20 + 23 + 26 + 29 + . . . + 50

Arithmetic Series

Teacher

Teacher

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48

Sigma Notation

Sigma ( ) is the Greek letter S.

And means the sum of the terms in a sequence.

The difference between Sn and sigma is that Sn is always the first to the nth term.

means start with 3rd and sum up through the 9th term.

Arithmetic Series

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49

Arithmetic Series

click

click

click

click

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27 Evaluate

Arithmetic Series

Teacher

Teacher

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28 Evaluate

Arithmetic Series

Teacher

Teacher

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29 Evaluate

Arithmetic Series

Teacher

Teacher

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53

How would evaluate ?

Arithmetic Series

Teacher

Teacher

There are 21 terms from the tenth term to the 30th. It is tempting to say 20 terms but you need to include the tenth term. If you don't believe this, write the term numbers out and count them. There are (high ­ low + 1) terms.

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54

30 Evaluate

Arithmetic Series

Teacher

Teacher

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55

Geometric Sequences

Return to Table of Contents

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A Geometric Sequence is a sequence in which the same number is multiplied to get from one term to the next. This common ratio is called r.

Find the next 3 terms in the geometric sequence

3, 6, 12, 24, . . .

5, 15, 45, 135, . . .

32, ­16, 8, ­4, . . .

16, 24, 36, 54, . . .

Geometric Sequence

Teacher

Teacher

If you don't know what number is being multiplies, divide the second term by the first.

PullPull

for extra hint

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31 Find the next term in geometric sequence: 6, ­12, 24, ­48, 96, . . .

Geometric Sequence

Teacher

Teacher

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32 Find the next term in geometric sequence: 64, 16, 4, 1, . . .

Geometric Sequence

Teacher

Teacher

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33 Find the next term in geometric sequence: 6, 15, 37.5, 93.75, . . .

Geometric Sequence

Teacher

Teacher

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34 Is the following sequence a geomtric one? 48, 24, 12, 8, 4, 2, 1

Yes

No

Geometric Sequence

Teacher

Teacher

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61

Geometric SequenceGeometric Sequences can be described by giving the

first term, a1, and the common ratio, r.

Examples: Find the first five terms of the geometric sequence described.

1) a1 = 6 and r = 3

2) a1 = 8 and r = ­.5

3) a1 = ­24 and r = 1.5

4) a1 = 12 and r = 2/3

Geometric Sequence

Teacher

Teacher

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35 Find the first four terms of the geometric sequence described: a1 = 6 and r = 4.

A 6, 24, 96, 384

B 4, 24, 144, 864

C 6, 10, 14, 18

D 4, 10, 16, 22

Geometric Sequence

Teacher

Teacher

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63

36 Find the first four terms of the geometric sequence described: a1 = 12 and r = ­1/2.

A 12, ­6, 3, ­.75

B 12, ­6, 3, ­1.5

C 6, ­3, 1.5, ­.75

D ­6, 3, ­1.5, .75

Geometric Sequence

Teacher

Teacher

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64

37 Find the first four terms of the geometric sequence described: a1 = 7 and r = ­2.

A 14, 28, 56, 112

B ­14, 28, ­56, 112

C 7, ­14, 28, ­56

D ­7, 14, ­28, 56

Geometric Sequence

Teacher

Teacher

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65

Geometric SequenceConsider the sequence: 3, 6, 12, 24, 48, 96, . . .To find the seventh term, just multiply the sixth term by 2.But what if I want to find the 20 th term?

Look for a pattern:a1 3

a2

a3

a4

a5

a6

a7

Do you see a pattern?

Geometric Sequence

click

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66

Geometric Sequence

Geometric Sequence

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Geometric SequenceFind the indicated term.

Example: a20 given a1 =3 and r = 2.

Example: a10 for 2187, 729, 243, 81

Geometric Sequence

Teacher

Teacher

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Geometric SequenceExample: Find r if a6 = .2 and a1 = 625

Example: Find n if a1 = 6, an = 98,304 and r = 4.

Geometric Sequence

Teacher

Teacher

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38 Find a12 in a geometric sequence wherea1 = 5 and r = 3.

Geometric Sequence

Teacher

Teacher

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70

39 Find a10 in a geometric sequence wherea1 = 7 and r = ­2.

Geometric Sequence

Teacher

Teacher

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40 Find a7 in a geometric sequence wherea1 = 10 and r = ­1/2.

Geometric Sequence

Teacher

Teacher

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72

41 Find r of a geometric sequence wherea1 = 3 and a10=59049.

Geometric Sequence

Teacher

Teacher

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73

42 Find n of a geometric sequence wherea1 = 72, r = .5, and an = 2.25

Geometric Sequence

Teacher

Teacher

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74

Geometric SequenceFind the missing term in the geometric sequence3, 9, 27,___

5, 1, 1/5 , ____

___, ­10, 50, ­250

­2, ___, ­32

Geometric Sequence

Teacher

Teacher

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75

Geometric Sequence

Find the missing terms in the geometric sequence.

5, ___, ___, 40

­54, ___, ___, 16

4, ___, ___, ___, 324

144, ___, ___, ___, ___, 4.5

Geometric Sequence

Teacher

Teacher

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76

43 What number(s) fill in the blanks of the geometric sequence: ___, 14, 98, 686

A ­14

B ­7

C ­2

D 2

E 7

F 8

G 10

H 12

I 28

J 50

Geometric Sequence

Teacher

Teacher

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77

44 What number(s) fill in the blanks of the geometric sequence: 5, ___, 80

A

C

E

G

I

J

H

F

D

B

Geometric Sequence

Teacher

Teacher

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78

45 What number(s) fill in the blanks of the geometric sequence: 4, ___, ___, ­500

A

C

E

G

I

J

H

F

D

B

Geometric Sequence

Teacher

Teacher

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79

Geometric Series

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80

The sum of a geometric series can be found using the formula:

Examples: Find Sn for each example.

1) a1= 5, r= 3, n= 6

2) a1= ­3, r= ­2, n=7

Geometric Series

Teacher

Teacher

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81

Sometimes information will be missing, so that using

isn't possible to start. Look to use to find missing information.Example: a1 = 16 and a5 = 243, find S5

Geometric Series

Teacher

Teacher

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82

Geometric Series

Teacher

Teacher

Example: a1 = 16 and a5 = 243, find S5 (continued)

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83

46 Find the indicated sum of the geometric series described: a1 = 10, n = 6, and r = 6

Geometric Series

Teacher

Teacher

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84

47 Find the indicated sum of the geometric series described: a1 = ­2, n = 5, and r = 1/4

Geometric Series

Teacher

Teacher

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85

48 Find the indicated sum of the geometric series described: a1 = 8, n = 6, and r = ­2

Geometric Series

Teacher

Teacher

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86

49 Find the indicated sum of the geometric series described: a1 = 8, n = 5, and a6 = 8192

Geometric Series

Teacher

Teacher

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87

50 Find the indicated sum of the geometric series described: r = 6, n = 4, and a4 = 2592

Geometric Series

Teacher

Teacher

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88

51 Find the indicated sum of the geometric series described: 8 ­ 12 + 18 ­ . . . find S7

Geometric Series

Teacher

Teacher

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89

Sigma ( )can be used to describe the sum of a geometric series.

Examples:

We can still use , but to do so we must examine the sigma notation.

n = 4 Why? The bounds on below and on top indicate that.

a1 = 6 Why? The coefficient is all that remains when the base is powered by 0.

r = 3 Why? In the exponential chapter this was our growth rate.

Geometric Series

Teacher

Teacher The bounds on below and on top indicate that.

The coefficient is all that remains when the base is powered by 0.

In the exponential chapter this was our growth rate.

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90

52 Find the sum:

Geometric Series

Teacher

Teacher

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91

53 Find the sum:

Geometric Series

Teacher

Teacher

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92

54 Find the sum:

Geometric Series

Teacher

Teacher

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93

InfiniteGeometric Series

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94

n Sn

1 3

2 12

3 39

4 120

5 363

6 1,092

7 3,279

8 9,840

n an

1 3

2 9

3 27

4 81

5 243

6 729

7 2,187

8 6,561

0 1 2 3 4 5 6 7 8 9

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

x

y

Because r = 3, the series isgrowing at increasing rate.

Infinite Geometric Series

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95

n Sn

1 64

2 96

3 112

4 120

5 124

6 126

7 127

8 127.5

n an

1 64

2 32

3 16

4 8

5 4

6 2

7 1

8 0.5

0 1 2 3 4 5 6 7 8 9

20

40

60

80

100

120

140

x

y

Because r = 1/2 , an ­­> 0.Notice Sn ­­> an asymptote?

Infinite Geometric Series

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96

n Sn

1 64

2 32

3 48

4 40

5 44

6 42

7 43

8 42.5

n an

1 64

2 ­32

3 16

4 ­8

5 4

6 ­2

7 1

8 ­0.5

0 1 2 3 4 5 6 7 8 9

­40

­20

20

40

60

80

100

120

140

x

y

Because r = ­1/2 , an ­­> 0.Notice Sn ­­> an asymptote?

Infinite Geometric Series

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97

For an infinite geometric series to approach a value,­1 < r < 1 then

The examples from the previous slides:

Example 1: a1 = 3 and r = 3.

Example 2: a1 = 64 and r= 1/2

Example 3: a1 = 64 and r= ­1/2

Infinite Geometric Series

Teacher

Teacher

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98

55 Find the sum of this infinite geometric series, if one exists:

Infinite Geometric Series

Teacher

Teacher

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99

56 Find the sum of this infinite geometric series, if one exists:

Infinite Geometric Series

Teacher

Teacher

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100

57 Find the sum of this infinite geometric series, if one exists:

Infinite Geometric Series

Teacher

Teacher

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101

58 Find the sum of this infinite geometric series, if one exists:

Infinite Geometric Series

Teacher

Teacher

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102

59 Find the sum of this infinite geometric series, if one exists:

Infinite Geometric Series

Teacher

Teacher

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103

60 Find the sum of this infinite geometric series, if one exists:

Infinite Geometric Series

Teacher

Teacher

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104

Special Sequences

Return to Table of Contents

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105

A recursive formula is one in which to find a term you need to know the preceding term.

So to know term 8 you need the value of term 7, and to know the nth term you need term n­1

In each example, find the first 5 terms

a1 = 6, an = an­1 +7 a1 =10, an = 4an­1 a1 = 12, an = 2an­1 +3

a1 a1

a2 a2

a3 a3

a4 a4

a5 a5

Special Sequences

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106

61 Find the first four terms of the sequence:

A 6, 3, 0, ­3

B 6, ­18, 54, ­162

C ­3, 3, 9, 15

D ­3, 18, 108, 648

a1 = 6 and an = an­1 ­ 3

Special Sequences

Teacher

Teacher

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107

62 Find the first four terms of the sequence:

A 6, 3, 0, ­3

B 6, ­18, 54, ­162

C ­3, 3, 9, 15

D ­3, 18, 108, 648

a1 = 6 and an = ­3an­1

Special Sequences

Teacher

Teacher

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108

63 Find the first four terms of the sequence:

A 6, ­22, 70, ­216

B 6, ­22, 70, ­214

C 6, ­14, 46, ­134

D 6, ­14, 46, ­142

a1 = 6 and an = ­3an­1 + 4

Special Sequences

Teacher

Teacher

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109

a1 = 6, an = an­1 +7 a1 =10, an = 4an­1 a1 = 12, an = 2an­1 +3

6 a1 10 a1 12

13 a2 40 a2 27

20 a3 160 a3 57

27 a4 640 a4 117

34 a5 2560 a5 237

The recursive formula in the first column represents an Arithmetic Sequence.We can write this formula so that we find a n directly.

Recall:

We will need a1 and d,they can be found both from the table and the recursive formula.

Special Sequences

Teacher

Teacher

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110

a1 = 6, an = an­1 +7 a1 =10, an = 4an­1 a1 = 12, an = 2an­1 +3

6 a1 10 a1 12

13 a2 40 a2 27

20 a3 160 a3 57

27 a4 640 a4 117

34 a5 2560 a5 237

The recursive formula in the second column represents a Geometric Sequence.We can write this formula so that we find an directly.

Recall:

We will need a1 and r,they can be found both from the table and the recursive formula.

Special Sequences

Teacher

Teacher

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111

a1 = 6, an = an­1 +7 a1 =10, an = 4an­1 a1 = 12, an = 2an­1 +3

6 a1 10 a1 12

13 a2 40 a2 27

20 a3 160 a3 57

27 a4 640 a4 117

34 a5 2560 a5 237

The recursive formula in the third column represents neither an Arithmetic or Geometric Sequence.

This observation comes from the formula where you have both multiply and add from one term to the next.

Special Sequences

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112

64 Identify the sequence as arithmetic, geometric,or neither.

A Arithmetic

B Geometric

C Neither

a1 = 12 , an = 2an­1 +7

Special Sequences

Teacher

Teacher

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113

65 Identify the sequence as arithmetic, geometric,or neither.

A Arithmetic

B Geometric

C Neither

a1 = 20 , an = 5an­1

Special Sequences

Teacher

Teacher

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114

66 Which equation could be used to find the nth term of the recursive formula directly?

A

B

C

D

a1 = 20 , an = 5an­1

an = 20 + (n­1)5

an = 20(5)n­1

an = 5 + (n­1)20

an = 5(20)n­1

Special Sequences

Teacher

Teacher

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115

67 Identify the sequence as arithmetic, geometric,or neither.

A Arithmetic

B Geometric

C Neither

a1 = ­12 , an = an­1 ­ 8

Special Sequences

Teacher

Teacher

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116

68 Which equation could be used to find the nth term of the recursive formula directly?

A

B

C

D

an = ­12 + (n­1)(­8)

an = ­12(­8)n­1

an = ­8 + (n­1)(­12)

an = ­8(­12)n­1

a1 = ­12 , an = an­1 ­ 8

Special Sequences

Teacher

Teacher

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117

69 Identify the sequence as arithmetic, geometric,or neither.

A Arithmetic

B Geometric

C Neither

a1 = ­12 , an = an­1 ­ 8a1 = 10 , an = an­1 + 8

Special Sequences

Teacher

Teacher

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118

70 Which equation could be used to find the nth term of the recursive formula directly?

A

B

C

D

an = 10 + (n­1)(8)

an = 10(8)n­1

an = 8 + (n­1)(10)

an = 8(10)n­1

a1 = 10 , an = an­1 + 8

Special Sequences

Teacher

Teacher

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119

71 Identify the sequence as arithmetic, geometric,or neither.

A Arithmetic

B Geometric

C Neither

a1 = 24 , an = (1/2)an­1

Special Sequences

Teacher

Teacher

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120

72 Which equation could be used to find the nth term of the recursive formula directly?

A

B

C

D

an = 24 + (n­1)(1/2)

an = 24(1/2)n­1

an = (1/2) + (n­1)24

an = (1/2)(24)n­1

a1 = 24 , an = (1/2)an­1

Special Sequences

Teacher

Teacher

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121

Special Recursive Sequences

Some recursive sequences not only rely on the preceding term, but on the two preceding terms.

Find the first five terms of the sequence:a1 = 4, a2 = 7, and an = an­1 + an­2

Special Sequences

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122

Special Recursive Sequences

Some recursive sequences not only rely on the preceding term, but on the two preceding terms.

Find the first five terms of the sequence:a1 = 6, a2 = 8, and an = 2an­1 + 3an­2

Special Sequences

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123

Special Recursive Sequences

Some recursive sequences not only rely on the preceding term, but on the two preceding terms.

Find the first five terms of the sequence:a1 = 10, a2 = 6, and an = 2an­1 ­ an­2

Special Sequences

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124

Special Recursive Sequences

Some recursive sequences not only rely on the preceding term, but on the two preceding terms.

Find the first five terms of the sequence:a1 = 1, a2 = 1, and an = an­1 + an­2

Special Sequences

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125

The sequence in the preceding example is called

The Fibonacci Sequence1, 1, 2, 3, 5, 8, 13, 21, . . .

where the first 2 terms are 1'sand any term there after is the sum of preceding two terms.

This is as famous as a sequence can get and is worth remembering.

Special Sequences

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126

73 Find the first four terms of sequence:

A

B

C

D

a1 = 5, a2 = 7, and an = a1 + a2

7, 5, 12, 19

5, 7, 35, 165

5, 7, 12, 195, 7, 13, 20

Special Sequences

Teacher

Teacher

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127

74 Find the first four terms of sequence:

A

B

C

D

a1 = 4, a2 = 12, and a n = 2an­1 ­ an­2

4, 12, ­4, ­20

4, 12, 4, 12

4, 12, 20, 284, 12, 20, 36

Special Sequences

Teacher

Teacher

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128

75 Find the first four terms of sequence:

A

B

C

D

a1 = 3, a2 = 3, and an = 3an­1 + an­2

3, 3, 6, 9

3, 3, 12, 39

3, 3, 12, 36

3, 3, 6, 21

Special Sequences

Teacher

Teacher

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129

Binomial Theorem

Return to Table of Contents

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130

Look for a pattern when powering a binomial.

Binomial Theorem

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131

One pattern comes from the coefficients.

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

This is Pascal's Triangle.

Binomial Theorem

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132

The coefficient of the rth term in the nth row can be found with:

From your study of Probability:• n! is the product of n and all of the natural numbers less than it.

Example: 4!= 4 x 3 x 2 x 1 = 240!=1

• the binomial coefficient formula is the same as the formula for combinations

Binomial Theorem

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133

Example: Find the coefficient of the 4 term of 6th power of a binomial.

Binomial Theorem

Teacher

Teacher

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134

76 Calculate

Binomial Theorem

Teacher

Teacher

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135

77 Calculate

Binomial Theorem

Teacher

Teacher

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136

78 Calculate

Binomial Theorem

Teacher

Teacher

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137

79 What is the binomial coefficient of the 4th term of x + y to the 8th power?

Binomial Theorem

Teacher

Teacher

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80 What is the binomial coefficient of the 2nd term of x + y to the 3rd power?

Binomial Theorem

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The second pattern of powering a binomial comes from the exponents.

Click to see the Binomial Expansion.

But what if instead of x + y we have 2x ­3?

Binomial Theorem

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The Binomial Theorem

Binomial Theorem

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Example: Expand (2x ­3)4

Binomial Theorem

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In general, to find the rth term of nth power of x + y

Binomial Theorem

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81 What is the coefficient of the 4th term (2x ­ 3) to the 5th power?

Binomial Theorem

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82 What is the coefficient of the 3rd term (2x ­ 3) to the 6th power?

Binomial Theorem

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Teacher

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83 What is the coefficient of the 6th term (2x ­ 3) to the 7th power?

Binomial Theorem

Teacher

Teacher