Top Banner
8.1 Sequences
21

8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Dec 27, 2015

Download

Documents

Magnus Barrett
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

8.1

Sequences

Page 2: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Quick Review

Let ( ) . Find the values of .4

1. (5)

2. (-1)

Evaluate the expression 1 for the given values of

, , and .

3. -2, 2, 3

4. 1, 2, 2

xf x f

xf

f

a n d

a n d

a n d

a n d

9

5

3

1

15

Page 3: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Quick Review

-1

2

2

0

lim

lim

Evaluate the expression for the given values of , , .

15. , 2, 3

26. 2, 1.5, 4

Find the value of the limit.

2 27.

4 1sin 4

8.

n

x

x

ar a r and n

a r n

a r n

x

x xx

x

2

75.6

2

1

4

Page 4: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence

Essential QuestionHow can we use calculus to define andevaluate sequences?

Page 5: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Defining a Sequence

1 2 3 1

1 2 3

A is a list of numbers written in an explicit order.

For example: , , ,..., ,... , where is the first

and is the of the sequence.

Let , , ,..., ,... be a funct

n

n n

n

n

a

a a a a a a

a

a a a a

sequence

term

nth term

1 2 3

ion with domain the set of positive

integers and range , , ,..., ,... . If the domain is finite, then

the sequence is a . If the domain is infinite, then

the sequence is an

na a a a

finite sequence

infinite .sequence

Page 6: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Defining a Sequence Explicitly1. Find the first four terms and the 100th term of the sequence {a

n} where

.

2

12

n

an

n Set n equal to 1, 2, 3, 4, and 100.

21

12

1

1

a3

1

22

12

2

2

a6

1

23

12

3

3

a11

1

24

12

4

4

a18

1

2100

12

100

100

a002,10

1

Page 7: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Defining a Sequence Recursively

2. Find the first three terms and the 7th term of the sequence defined recursively by the conditions: b

1 = 4 and b

n = b

n – 1 – 2 for all n > 2.

41 b

2122 bb 21 b 24 2

2133 bb 22 b 22 0

2177 bb 26 b 26 8

Page 8: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Arithmetic SequenceA sequence {a} is an arithmetic sequence if it can be written in the form {a, a + d, a + 2d, . . . , a + (n – 1)d, . . .} for some constant d. The number d is the common difference.

.2 allfor 1 ndaa nn

Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d:

Page 9: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Defining Arithmetic Sequences3. Given the arithmetic sequence: – 3, 1, 5, 9, . . . find

a. the common difference,

b. the ninth term,

c. a recursive rule for the nth term,

d. an explicit rule for the nth term.

. is differencecommon The a. 12 aa 31 4 dnaan 1 b. 1

9a 3 19 4 29:is rule recursive The c. ,31 a 41 nn aa

dnaan 1 :is ruleexplicit The d. 1

na 3 1 n 4 74 n

Page 10: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Geometric SequenceA sequence {a} is an geometric sequence if it can be written in the

form {a, a . r, a . r2, . . . , a . r n – 1 , . . .} for some nonzero constant r. The number r is the common ratio.

.2 allfor 1 nraa nn

Each term in an geometric sequence can be obtained recursively from its preceding term by multiplying by r:

Page 11: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Defining Geometric Sequences4. Given the geometric sequence: 1, – 3, 9, – 27, . . . find

a. the common ratio,

b. the tenth term,

c. a recursive rule for the nth term,

d. an explicit rule for the nth term.

. is ratiocommon The a.1

2

a

a

1

3 31

1 b. nn raa

9a 1 3 110 683,19:is rule recursive The c. ,11 a 13 nn aa

11 :is ruleexplicit The d. n

n raa

na 1 3 1n 13 n

Page 12: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Constructing a Sequence5. The second and fifth term of a geometric sequence are – 6 and 48,

respectively. Find the first term, common ratio and an explicit rule for the nth term.

121

151

ra

ra1

1

41

ra

ra

6

48

3r 82r

61 ra

62 1 a

31 a1

1 :is ruleexplicit The nn raa

na 3 2 1n 11 231 nn

Page 13: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Graphing a Sequence Using Parametric Mode

6. Draw a graph of the sequence {an} with . . . 3, 2, ,1 ,1

1

nn

na n

n

Change the mode on your calculator to parametric and dot.

T

TT 11Y T,XLet 1T1T

1T ,20T ,1T stepmaxmin Set your window for the following:

2X ,20X ,0X sclmaxmin 1Y ,2Y ,2Y sclmaxmin

Page 14: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Graphing a Sequence Using Sequence Graphing Mode

7. Graph the sequence defined recursively by b1 = 4 and b

n = b

n – 1 + 2 for

all n > 2.Change the mode on your calculator to sequence and dot.Replace b

n by u(n).

Select nMin = 1, u(n) =u(n – 1) + 2, and u(nMin) = {4}.

Page 15: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Graphing a Sequence Using Sequence Graphing Mode

7. Graph the sequence defined recursively by b1 = 4 and b

n = b

n – 1 + 2 for

all n > 2.

Set nMin = 1, uMax = 10, PlotStart = 1, PlotStep = 1, and graph in the [0, 10] by [– 5, 25] viewing window.

Page 16: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Limit Let L be a real number. The sequence a has limit L as n approaches ∞

if, given any positive number , there is a positive number M such that for all n > M we have . Lan

We write Lann

lim and say that the sequence converges to L.

Sequences that do not have limits diverge.

Page 17: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Properties of Limits If L and M are real numbers and

1. Sum Rule:

Lann

lim and , lim Mbn

n

then

MLba nnn

lim

2. Difference Rule: MLba nnn

lim

3. Product Rule: MLba nnn

lim

4. Constant Multiple Rule: Lccann

lim

5. Quotient Rule: 0,lim

MM

L

b

a

n

n

n

Page 18: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Finding the Limit of a Sequence8. Determine whether the sequence converges or diverges. If it

converges, find its limit.

n

nan

12

Graph it, changing the mode to parametric and dot.

Find the limit analytically, using the Properties of Limits:

n

nn

12lim

nn

12lim

nnn

1lim2lim

2 0 2

Page 19: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

The Sandwich Theorem for Sequencesand if there is an integer N for whichLca n

nn

n

lim lim If

LbNncba nn

nnn

lim then , allfor

Absolute Value Theorem .0 lim then ,0 lim If . sequence heConsider t

nn

nn

n aaa

Page 20: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Example Using the Sandwich Theorem to find the Limit of a Sequence

9. Show that the following sequence converges and find its limit.

n

nan

cos 1cos x

n

ncos n

ncos

n

1

n

1

n

ncos

n

1

nn

1lim 0

nn

1lim 0

n

nn

coslim 0

Page 21: 8.1 Sequences Quick Review What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence.

Pg. 441, 8.1 #1-43 odd