Copyright © 2007 Pearson Education, Inc. Slide 8-1 Warm-Up Find the next term in the sequence: 1, 1, 2, 6, 24, 120,…

Copyright © 2007 Pearson Education, Inc. Slide 8-2

Warm-Up• Find the first four

terms of the sequence given by 3 2na n

1 13 2 1a

2 23 2 4a

3 33 2 7a

4 3 104 2a

Find the first four terms of the sequence given by 3 ( 1)n

na

11 3 ( 1) 3 1 2a

22 3 ( 1) 3 1 4a

33 3 ( 1) 3 1 2a

44 3 ( 1) 3 1 4a


Chapter 8: Sequences and Series

2015


Chapter 8: Sequences, Series, and Probability

8.1 Sequences and Series

8.2 Arithmetic Sequences and Series

8.3 Geometric Sequences and Series

8.4 Mathematical Induction

8.5 The Binomial Theorem


8.1 Sequences

• f (x) notation is not used for sequences.• Write • Sequences are written as ordered lists

• a1 is the first element, a2 the second element, and so on

A sequence is a function that has a set of natural numbers as its domain.

( )na f n

1 2 3, , , ...a a a


8.1 Sequences

A sequence is often specified by giving a formula forthe general term or nth term, an.

Example Find the first four terms for the sequence

Solution

1

2n

na

n

1 2 / 3,a 3 4 / 5,a 2 3 / 4,a 4 5 / 6a


8.1 Graphing Sequences

The graph of a sequence, an, is the graph of thediscrete points (n, an) for n = 1, 2, 3, …

Example Graph the sequence an = 2n.

Solution


8.1 Sequences

• A finite sequence has domain the finite set {1, 2, 3, …, n} for some natural number n.Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

• An infinite sequence has domain {1, 2, 3, …}, the set of all natural numbers.

Example 1, 2, 4, 8, 16, 32, …


8.1 Convergent and Divergent Sequences

• A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number.

• A sequence that is not convergent is said to be divergent.



Example :

Find the first 5 terms of the sequence .

Is the sequence convergent or divergent?

1na

n



Solution: The sequence converges to 0.

The terms of the sequence 1, 0.5, 0.33.., 0.25, … grow smaller and smaller approaching 0. This can be seen graphically.

1na

n



Example :

Find the first 6 terms of the sequence .

Is the sequence convergent or divergent?

2na n



Solution: The sequence is divergent.The terms grow large without bound

1, 4, 9, 16, 25, 36, 49, 64, …

and do not approach any one number.

2na n



Example Is the sequence convergent or divergent?

2

2

2 3

3 2 5n

n na

n n

Solution: The sequence converges to 2/3


Finding Terms of a Sequence• Write out the first five terms of the sequence given by

( 1)

2 1

n

nan

Solution:

1

1 1

( 1) 11

2 1 2 1a

2

2 2

( 1) 1 1

2 1 4 1 3a

3

3

( 1) 1 1

2 1 63 1 5a

4

4 4

( 1) 1 1

2 1 8 1 7a

5

5

( 1) 1 1

2 1 105 1 9a


Finding the nth term of a Sequence

• Write an expression for the apparent nth term (an) of each sequence.

• a. 1, 3, 5, 7, … b. 2, 5, 10, 17, …Solution:

a. n: 1 2 3 4 . . . n

terms: 1 3 5 7 . . . an Apparent pattern: Each term

is 1 less than twice n, which implies that 2 1na n

b. n: 1 2 3 4 … n

terms: 2 5 10 17 … an

Apparent pattern: Each term is 1 more than the square of n, which implies that 2 1na n


Additional Example• Write an expression for the apparent nth term of the sequence:

2 3 4 5, , , ,...

1 2 3 4

Solution: : 1 2 3 4 ...

2 3 4 5 terms: ...

1 2 3 4 n

n n

a

Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that

1n

na

n


Factorial Notation• If n is a positive integer, n factorial is defined by

As a special case, zero factorial is defined as 0! = 1. Here are some values of n! for the first several nonnegative integers.

Notice that 0! is 1 by definition.

! 1 2 3 4... ( 1)n n n

0! 1

1! 12! 1 2 2

3! 1 2 3 6 4! 1 2 3 4 24 5! 1 2 3 4 5 120

The value of n does not have to be very large before the value of n! becomes huge. For instance, 10! = 3,628,800.


Finding the Terms of a Sequence Involving Factorials

• List the first five terms of the sequence given by

Begin with n = 0.

2

!

n

nan

0

0

2 11

0! 1a

1

1

2 22

1! 1a

2

2

2 42

2! 2a

3

3

2 8 4

3! 6 3a

4

4

2 16 2

4! 24 3a


Evaluating Factorial Expressions• Evaluate each factorial expression. Make sure you use parentheses when

necessary. a. b. c. 8!

2! 6!

2! 6!

3! 5!

!

( 1)!

n

n

a.

b.

c.

8! 1 2 3 4 5 6 7 8 7 828

2! 6! 1 2 1 2 3 4 5 6 2

2! 6! 1 2 1 2 3 4 5 6 62

3! 5! 1 2 3 1 2 3 4 5 3

! 1 2 3...( 1)

( 1)! 1 2 3...( 1)

n n nn

n n


Additional Example• Write an expression for the apparent nth term of the sequence:

2 3 4 52 2 2 21,2, , , , ,...

2 6 24 120

Solution:

2 3 4 5

: 1 2 3 4 5 6 ...

2 2 2 2 terms: 1, 2, , , , ...

2 6 24 12

0 n

n n

a

Apparent pattern: Each term has a numerator that is 1 greater than its denominator, which implies that

12

1 !

n

nan


Have you ever seen this sequence before?

• 1, 1, 2, 3, 5, 8 …• Can you find the next three terms in the

sequence?• Hint: 13, • 21, 34• Can you explain this pattern?


The Fibonacci Sequence• Some sequences are defined recursively. To define a sequence

recursively, you need to be given one or more of the first few terms. A well-known example is the Fibonacci Sequence.

• The Fibonacci Sequence is defined as follows:

0 1 2 11, 1, , where 2k k ka a a a a k

Write the first six terms of the Fibonacci Sequence:

1 1a 0 1a

2 1 02 2 1 1 1 2a a a a

2 1 13 3 2 1 2 3a a a a

2 1 24 4 3 2 3 5a a a a

2 1 35 5 4 3 5 8a a a a


Example

• Write the first five terms of the recursively defined sequence:

1 15, 3k ka a a

Solution: 5, 8, 11, 14, 17


Homework

• Day 1: Pg. 563 1-9odd, 21-23odd, 35-69 odd

• Day 2: 71-81 odd, 91-103 odd


HWQWrite an expression for the apparent nth term of the sequence.

1 3 7 15 311 ,1 ,1 ,1 ,1 ,...

2 4 8 16 32


8.1 Day 2Series 2015


Summation Notation

• Definition of Summation NotationThe sum of the first n terms of a sequence is represented by

Where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation.

1 2 3 41

...n

n ni

a a a a a a


8.1 Series and Summation Notation

A finite series is an expression of the form

and an infinite series is an expression of the form

.

1 2 31

...n

n n ii

S a a a a a

1 2 31

... ...n ii

S a a a a a


Summation Notation for Sums• Find each sum.a. b. c. 5

1

3i

i

62

3

(1 )k

k

8

0

1

!i i

Solution:

a. 5

1

3 3(1) 3(2) 3(3) 3(4) 3(5)

3(1 2 3 4 5) or 3 6 9 12 15

45

i

i


Solutions continuedb.

c.

62 2 2 2 2

3

(1 ) (1 3 ) (1 4 ) (1 5 ) (1 6 )

10 17 26 37

90

k

k

8

0

1 1 1 1 1 1 1 1 1 1

! 0! 1! 2! 3! 4! 5! 6! 7! 8!

1 1 1 1 1 1 1 1 1

2 6 24 120 720 5040 40320 2.71828

i i

Notice that this summation is very close to the irrational number . It can be shown that as more terms of the sequence whose nth term is 1/n! are added, the sum becomes closer and closer to e.

2.718281828e



Summation Properties

If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then for every positive integer n,

(a) (b)

(c)

1

n

i

c nc

1 1

n n

i ii i

ca c a

1 1 1

( )n n n

i i i ii i i

a b a b



Summation Rules

1

2 2 2 2

1

2 23 3 3 3

1

( 1)1 2 ...

2

( 1)(2 1)1 2 ...

6

( 1)1 2 ...

4

n

i

n

i

n

i

n ni n

n n ni n

n ni n

These summation rules can be proven by mathematical induction.



Example Use the summation properties to evaluate (a) (b) (c)

Solution

(a)

40

1

5i

22

1

2i

i

142

1

(2 3)i

i

40

1

5 40(5) 200i



(b)

(c)

14 14 14 14 142 2 2

1 1 1 1 1

(2 3) 2 3 2 3

14(14 1)(2 14 1)2 14(3) 1988

6

i i i i i

i i i

22 22

1 1

22(22 1)2 2 2 506

2i i

i i

22

1

2i

i

142

1

(2 3)i

i

(b) (c)


Homework

• Day 1: Pg. 563 1-9odd, 21-23odd, 35-69 odd

• Day 2: 71-81 odd, 91-103 odd

Documents

Copyright © 2007 Pearson Education, Inc. Slide 8-1 Warm-Up Find the next term in the sequence: 1, 1, 2, 6, 24, 120,…