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

PreCalculus

Sequences and Series

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20150324

Table of Contents

Arithmetic Sequences

Arithmetic Series

Geometric Sequences

Geometric Series

Special Sequences

Infinite Geometric Series

Binomial Theorem

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Arithmetic Sequences

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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

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

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

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

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

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

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

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

A 3, 7, 10, 14

B 4, 7, 11, 13

C 3, 7, 11, 15

D 3, 1, 5, 9

Arithmetic Sequence

Teacher

A 7, 10, 13, 16

B 4, 7, 10, 13

C 1, 4, 7,10

D 3, 5, 7, 9

Arithmetic Sequence

Teacher

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

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

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

value?

Arithmetic Sequence

Teacher

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

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

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

30 = a1 + (14)7 30 = a1 + 98 58 = a1

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

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

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

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

Arithmetic Sequence

Teacher

10 Find a17 when a1 = 12 and d = 0.5

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

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

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

Arithmetic Sequence

Find the missing terms

2, ___ , ___ , 23

4, ___ , ___ , 14

7, ___ , ___, ___, 39

9, ___ , ___ , ___, ___ , 34

Arithmetic Sequence

Teacher

15 Find the missing term: 4, ___ , 16

Arithmetic Sequence

Teacher

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

A 6, 2

B 6, 2

C 5, 1

D 4, 2

Arithmetic Sequence

Teacher

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

A 27, 51

B 33, 54

C 37, 51

D 34, 58

Arithmetic Sequence

Teacher

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

Arithmetic Sequence

Teacher

Arithmetic Series

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

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

Arithmetic Series

Teacher

Arithmetic Series

Teacher

Arithmetic Series

Teacher

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

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

Find Sn for each arithmetic series.

Example: a1 = 7 and a12 = 23

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

Arithmetic Series

Teacher

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

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

Arithmetic Series

Teacher

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

Arithmetic Series

Teacher

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

Arithmetic Series

Teacher

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

Arithmetic Series

Teacher

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

Arithmetic Series

Teacher

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

Arithmetic Series

Teacher

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

27 Evaluate

Arithmetic Series

Teacher

28 Evaluate

Arithmetic Series

Teacher

29 Evaluate

Arithmetic Series

Teacher

How would evaluate ?

Arithmetic Series

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.

30 Evaluate

Arithmetic Series

Teacher

Geometric Sequences

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

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

PullPull

for extra hint

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

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

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

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

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

Do you see a pattern?

Geometric Sequence

Geometric SequenceFind the indicated term.

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

Example: a10 for 2187, 729, 243, 81

Geometric Sequence

Teacher

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

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

5, 1, 1/5 , ____

___, 10, 50, 250

2, ___, 32

Geometric Sequence

Teacher

Geometric Sequence

Find the missing terms in the geometric sequence.

5, ___, ___, 40

54, ___, ___, 16

4, ___, ___, ___, 324

144, ___, ___, ___, ___, 4.5

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

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

Geometric Sequence

Teacher

Geometric Series

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

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

Geometric Series

Teacher

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

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

Geometric Series

Teacher

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

Geometric Series

Teacher

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

Geometric Series

Teacher

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

Geometric Series

Teacher

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

Geometric Series

Teacher

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

Geometric Series

Teacher

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.

52 Find the sum:

Geometric Series

Teacher

53 Find the sum:

Geometric Series

Teacher

54 Find the sum:

Geometric Series

Teacher

InfiniteGeometric Series

6 1,092

7 3,279

8 9,840

7 2,187

8 6,561

0 1 2 3 4 5 6 7 8 9

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

8 127.5

0 1 2 3 4 5 6 7 8 9

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

8 42.5

0 1 2 3 4 5 6 7 8 9

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

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

Teacher

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

Teacher

Special Sequences

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

Special Sequences

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

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

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

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

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

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

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

A Arithmetic

B Geometric

C Neither

a1 = 12 , an = 2an1 +7

Special Sequences

Teacher

A Arithmetic

B Geometric

C Neither

a1 = 20 , an = 5an1

Special Sequences

Teacher

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

a1 = 20 , an = 5an1

an = 20 + (n1)5

an = 20(5)n1

an = 5 + (n1)20

an = 5(20)n1

Special Sequences

Teacher

A Arithmetic

B Geometric

C Neither

a1 = 12 , an = an1 8

Special Sequences

Teacher

an = 12 + (n1)(8)

an = 12(8)n1

an = 8 + (n1)(12)

an = 8(12)n1

a1 = 12 , an = an1 8

Special Sequences

Teacher

A Arithmetic

B Geometric

C Neither

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

Special Sequences

Teacher

an = 10 + (n1)(8)

an = 10(8)n1

an = 8 + (n1)(10)

an = 8(10)n1

a1 = 10 , an = an1 + 8

Special Sequences

Teacher

A Arithmetic

B Geometric

C Neither

a1 = 24 , an = (1/2)an1

Special Sequences

Teacher

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

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

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

Special Sequences

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

Special Sequences

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

Special Sequences

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

73 Find the first four terms of sequence:

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

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

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

Binomial Theorem

Look for a pattern when powering a binomial.

Binomial Theorem

One pattern comes from the coefficients.

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

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

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

Binomial Theorem

Teacher

76 Calculate

Binomial Theorem

Teacher

77 Calculate

Binomial Theorem

Teacher

78 Calculate

Binomial Theorem

Teacher

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

Binomial Theorem

Teacher

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

Binomial Theorem

Teacher

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

The Binomial Theorem

Binomial Theorem

Example: Expand (2x 3)4

Binomial Theorem

Teacher

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

Binomial Theorem

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

Binomial Theorem

Teacher

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

Binomial Theorem

Teacher

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

Binomial Theorem

Teacher

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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