Chapter 6

Sequences And Series

Look at these number sequences carefully can you guess the next 2 numbers?

What about guess the rule?

30 40 50 60 70 80

17 20 292623 32

---------------------------------------------------------------------------------------------------------------------

---------------------------------------------------------------------------------------------------------------------

48 41 34 27 20 13

+10

+3

-7

Can you work out the missing numbers in each of these sequences?

50

30

17515012510075

---------------------------------------------------------------------------------------------------------------------

50 70 90 110 130

---------------------------------------------------------------------------------------------------------------------

171176181186191196

---------------------------------------------------------------------------------------------------------------------

256266276286296306

+25

+20

-5

-10

Now try these sequences – think carefully and guess the last number!

1 2 164 7 11

3

---------------------------------------------------------------------------------------------------------------------

12 24 48 966

---------------------------------------------------------------------------------------------------------------------

0.5 2 3.5 5 6.5 8

---------------------------------------------------------------------------------------------------------------------

7 -5-214 -8

+1, +2, +3 …

double

+ 1.5

-3

This is a really famous number sequence which was discovered by an Italian mathematician a long time ago.

It is called the Fibonacci sequence and can be seen in many natural things like pine cones and sunflowers!!!0 1 1 2 3 5 8 13 21 etc…Can you see how it is made? What will the next number be?

34!

Guess my rule!

For these sequences I have done 2 maths functions!

3 7 3115 63 127

2x -1

2 3317953

2x +1

What is a Number Sequence?

A list of numbers where there is a pattern is called a number

sequenceThe numbers in the sequence are

said to be its members or its terms.

SequencesSequences

To write the terms of a sequence given the nth term

Given the expression: 2n + 3, write the first 5 terms

In this expression the letter n represents the term number. So, if we substitute the term number for the letter n we will find value that particular term.

The first 5 terms of the sequence will be using values for n of: 1, 2, 3, 4 and 5

term 12 x 1 + 3

5

term 22 x 2 + 3

7

term 32 x 3 + 3

9

term 42 x 4 + 3

11

term 52 x 5 + 3

13

SequencesSequencesNow try these:

Write the first 3 terms of these sequences:

1) n + 2

2) 2n + 5

3) 3n - 2

4) 5n + 3

5) -4n + 10

6) n2 + 2

3, 4, 5

7, 9, 11

1, 4, 7

8, 13, 18

6, 2, - 2,

3, 6, 11,

6B - The General Term of A Number Sequence

Sequences may be defined in one of the following ways:

• listing the first few terms and assuming the pattern represented continues indefinitely

• giving a description in words

• using a formula which represents the general term or nth term.

The first row has three bricks, the second row has four bricks, and the third row has five bricks.

• If un represents the number of bricks in row n (from the top) then u1 = 3, u2 = 4, u3 = 5, u4 = 6, ....

This sequence can be describe in one of four ways:

• Listing the terms:

• u1 = 3, u2 = 4, u3 = 5, u4 = 6, ....

This sequence can be describe in one of four ways:

• Using Words: The first row has three bricks and each successive row under the row has one more brick...

This sequence can be describe in one of four ways:

• Using an explicit formula: un = n + 2u1 = 1 + 2 = 3u2 = 2 + 2 = 4u3 = 3 + 2 = 5u4 = 4 + 2 = 6, ....

This sequence can be describe in one of four ways:

• Using a graph

What you really need to know!

An arithmetic sequence is a sequence in which the difference between any two consecutive terms, called the common difference, is the same. In the sequence 2, 9, 16, 23, 30, . . . , the common difference is 7.

What you really need to know!

A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256, . . , the common ratio is 4.

Example 1:

State whether the sequence -5, -1, 3, 7, 11, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2:

SubtractSubtract Common differenceCommon difference

11 – 7 11 – 7 447 – 3 7 – 3 443 – -1 3 – -1 44

-1 – -5 -1 – -5 44

-5, -1, 3, 7, 11,

Arithmetic! + 4

15, 19, 23

Example 2:

State whether the sequence 0, 2, 6, 12, 20, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2:

SubtractSubtract Common differenceCommon difference

20 – 12 20 – 12 8812 – 6 12 – 6 666 – 2 6 – 2 442 – 0 2 – 0 22

0, 2, 6, 12, 20 …

Not Arithmetic!

Example 3:

State whether the sequence 2, 4, 4, 8, 8, 16, 16 … is geometric. If it is, state the common ratio and write the next three terms.

Example 3:

DivideDivide Common ratioCommon ratio

16 16 ÷÷ 16 16 11

16 16 ÷÷ 8 8 22

8 8 ÷÷ 8 8 11

8 8 ÷÷ 4 4 22

4 4 ÷÷ 4 4 11

4 4 ÷÷ 2 2 22

2, 4, 4, 8, 8, 16, 16, …

Not Geometric!

Example 4:

State whether the sequence 27, -9, 3, -1, 1/3, … is geometric. If it is, state the common ratio and write the next three terms.

Example 4:

DivideDivide Common ratioCommon ratio

1/3 1/3 ÷÷ -1 -1 -1/3-1/3-1 -1 ÷÷ 3 3 -1/3-1/33 3 ÷÷ -9 -9 -1/3-1/3

-9 -9 ÷÷ 27 27 -1/3-1/3

27, -9, 3, -1, 1/3,

Geometric! • -1/3

-1/9, 1/27, -1/81

ClassworkClassworkPage 154 (6B) All 5. Page 154 (6B) All 5.

HomeworkHomeworkCompare Arithmetic Compare Arithmetic

and Geometric and Geometric SequencesSequences

An An Arithmetic Arithmetic SequenceSequence is definedis defined as as

a sequence in which a sequence in which there is a there is a common common differencedifference between between consecutive terms.consecutive terms.

Which of the following sequences are arithmetic?

Identify the common difference.

3, 1, 1, 3, 5, 7, 9, . . .

15.5, 14, 12.5, 11, 9.5, 8, . . .

84, 80, 74, 66, 56, 44, . . .

8, 6, 4, 2, 0, . . .

50, 44, 38, 32, 26, . . .

YES 2d

YES

YES

NO

NO

1.5d

6d

The common

difference is

always the

difference between

any term and the

term that proceeds

that term.26, 21, 16, 11, 6, . . .

Common Difference = 5

The general form of an ARITHMETIC sequence.

1uFirst Term:

Second Term: 2 1 1u u d

Third Term:

Fourth Term:Fifth Term:

3 1 2u u d

4 1 3u u d

5 1 4u u d

nth Term: 1 1na a dn

Formula for the nth term of an ARITHMETIC sequence.

1 1nu u n d

nu th The n term

The term numbern

The common differenced

1u The 1st term

If we know any

If we know any threethree of these of these we ought to be

we ought to be able to find the

able to find the fourth.fourth.

Given: 79, 75, 71, 67, 63, . . .Find: 32u

1 79

4

32

u

d

n

IDENTIFY SOLVE

1 ( 1)nu u n d

32 79 (32 1)( 4)u

32 45u

Given: 79, 75, 71, 67, 63, . . .

Find: What term number is (-169)?

1 79

4

169n

u

d

u

IDENTIFY SOLVE

1 ( 1)nu u n d

)4)(1(79169 n

63nIf it’s not an integer, it’s not a term in the

sequence

Given:10

12

3.25

4.25

u

u

1

3

3.25

4.25

3

u

u

n

1 1

4.25 3.25 3 1

0.5

nu u n d

d

d

IDENTIFY SOLVE

Find: 1u

What’s the real question? The Difference

Given:10

12

3.25

4.25

u

u

10 3.25

0.5

10

u

d

n

1

1

1

1

3.25 10 1 0.5

1.25

nu u n d

u

u

IDENTIFY SOLVE

Find: 1u

Homework

Page 156 2 - 11( Any 8 Problems)

Take Home Test Due Tuesday.

GeometricGeometric SeriesSeries

Geometric SequenceGeometric Sequence

• The ratio of a term to it’s The ratio of a term to it’s previous term is constant.previous term is constant.

• This means you multiply by the This means you multiply by the same number to get each term.same number to get each term.

• This number that you multiply by This number that you multiply by is called the is called the common ratiocommon ratio ( (rr).).

ExampleExample: Decide whether each : Decide whether each sequence is geometric.sequence is geometric.

• 4,-8,16,-32,…

• -8/4=-2

• 16/-8=-2

• -32/16=-2

• Geometric (common ratio is -2)

• 3,9,-27,-81,243,…

• 9/3=3

• -27/9=-3

• -81/-27=3

• 243/-81=-3

• Not geometric

Rule for a Geometric Rule for a Geometric SequenceSequenceuunn=u=u11r r n-1n-1

ExampleExample: Write a rule for the nth : Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,term of the sequence 5, 2, 0.8, 0.32,

… . Then find u… . Then find u88..

•First, find r.First, find r.

•r= r= 22//5 5 = .4= .4

•uunn=5(.4)=5(.4)n-1n-1

uu88=5(.4)=5(.4)8-18-1

uu88=5(.4)=5(.4)77

uu88=5(.0016384)=5(.0016384)

uu88=.008192=.008192

One term of a geometric sequence is One term of a geometric sequence is uu4 4 = 3= 3. .

The common ratio is The common ratio is r = 3r = 3. Write a rule for the . Write a rule for the nth nth term. Then graph the sequence.term. Then graph the sequence.

• If uIf u44=3, then when =3, then when n=4, un=4, unn=3.=3.

• Use uUse unn=u=u11rrn-1n-1

3=u3=u11(3)(3)4-14-1

3=u3=u11(3)(3)33

3=u3=u11(27)(27)11//99=a=a11

• uunn=u=u11rrn-1n-1

uunn=(=(11//99)(3))(3)n-1n-1

• To graph, graph To graph, graph the points of the points of the form (n,uthe form (n,unn).).

• Such as, (1,Such as, (1,11//99), ), (2,(2,11//33), (3,1), ), (3,1), (4,3),…(4,3),…

Two terms of a geometric sequence are Two terms of a geometric sequence are uu22= -4= -4

and and uu66= -1024= -1024. Write a rule for the . Write a rule for the nthnth term. term.• Write 2 equations, one for each given term.

u2 = u1r2-1 OR -4 = u1r

u6 = u1r6-1 OR -1024 = u1r5

• Use these equations & sub in to solve for u1 & r.-4/r=u1

-1024=(-4/r)r5

-1024 = -4r4

256 = r4

4 = r & -4 = r

If r = 4, then u1 = -1.

un=(-1)(4)n-1

If r = -4, then u1 = 1.

un=(1)(-4)n-1

un=(-4)n-1

Both Both Work!Work!

6D1 (4 a and b)6D1 (4 a and b)

• 5, 10, 20, 40

• So, geometric sequence with u1 = 5 r = 2

10 20 402

5 10 20

11

nnu u r

15 2nnu x

1415 5 2 81,920u x

6D1 (9a)6D1 (9a)

• u4 = 24

u7 = 192

611 3

192

24

u r

u r

31 1 24u u xr

61 1 192u u xr

3 8 2r r 3

1

1

1

2 24

8 24

3

u

u

u

13 2nnu x

Homework

• Page 160 (6D.1 All)

• Take Home Test Due Tuesday

Compound Interest

8 - 47

100110

121

1000

1210

1331

1100100 100

110

Time(Years)0 1 2 3 4

Amount $1000

Amount $1000

110

InterestInterestInterestInterest

100

InterestInterest

133.1

Compounding Period

Compounding Period

Compounding Period

Compounding Period

InterestInterest

121

Compound Interest- Future Value

COMPOUND INTEREST FORMULA

tn

nr

1 PV FV

Where Where FVFV is the is the Future Value Future Value in in tt yearsyears and and PVPV is the is the Present Value Present Value amount amount started with at an annual interest rate started with at an annual interest rate rr compounded compounded nn times per year. times per year.

Find the amount that results from the Find the amount that results from the investment:investment:

$50 invested at 6% compounded monthly $50 invested at 6% compounded monthly after a period of 3 years.after a period of 3 years.

EXAMPLEEXAMPLE

)3(12

12.06

1 50 FV $59.83$59.83

Investing $1,000 at a rate of 10% Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, compounded annually, quarterly, monthly, and daily will yield the following amounts and daily will yield the following amounts after 1 year:after 1 year:

FV = PV(1 + r) = 1,000(1 + .1) = $1100.00FV = PV(1 + r) = 1,000(1 + .1) = $1100.00

COMPARING COMPARING COMPOUNDING PERIODSCOMPOUNDING PERIODS

$1103.81 4.1

1 1000 FV

4

Investing $1,000 at a rate of 10% Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, compounded annually, quarterly, monthly, and daily will yield the following amounts and daily will yield the following amounts after 1 year:after 1 year:

COMPARING COMPARING COMPOUNDING PERIODSCOMPOUNDING PERIODS

$1104.71 12.1

1 1000 FV

12

$1105.16 365.1

1 1000 FV

365

Investing $1,000 at a rate of 10% Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, compounded annually, quarterly, monthly, and daily will yield the following amounts and daily will yield the following amounts after 1 year:after 1 year:

Interest EarnedInterest Earned

$ 1 0 4 . 7 1 $ 1 0 0 0- $ 1 1 0 4 . 7 1 E a r n e d I n t

$105.16 $1000- $1105.16 I.E .

Page 165 6D.3, #1 a & b

1 1( )nnu u r

1 3000 1.1 3u r n

34 3000(1.1) 3993u

The investment will amount to $3993

B.) Interest = amount after 3 yrs – initial amount

$3993 - $3000 = $993

Page 165 6D.3, #2 a & b

1 1( )nnu u r

1 20,000 1.12 4u r n

45 20000(1.12) 31470.39u

The investment will amount to €31470.39B.) Interest = €31470.39 – €31470.39

€11470.39

Page 165. 6D.3Page 165. 6D.3

(3 – 10). (3 – 10).

HomeworkHomework

6E – Sigma Notation

Vocabulary

k

nna

1

maximum valueof n

starting valueof n

expression forgeneral term

Sigma – “take the

sum of…”

Read: “the summation from n = 1 to k of an”

Introduction to Sigma Notation

3 + 6 + 9 + 12 + 15 = ∑ 3n5

n = 1

5

n = 1∑ 3n

Is read as “the sum from n equals 1 to 5 of 3n.”

index of summation lower limit of summation

upper limit of summationHow many terms given?

58

Formulas

Arithmetic Sum:

1

2nn a a

S

1 First Term

Last Term

Amount of Termsn

a

a

n

59

Formulas

Sigma Form of Arithmetic Series:

11

1k

n

a d n

1

Amount of Terms

First Term

Common Difference

Variable

k

a

d

n

04/19/23 20:21 60

Example 1

Write in Sigma Notation, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

11

1k

n

a d n

11

1k

n

a d n

1

Amount of Terms

First Term

Common Difference

Variable

k

a

d

n

12

n

1 1a d n

1 2 1

1 2 2

2 1

n

n

n

11

2 1n 11

1

2 1n

n

61

11

Example 2

Write in Sigma Notation, 26 + 23 + 20 + 17 + 14 + 11 + 8 + 5

8

1

3 29n

n

04/19/23 20:21 62

Your Turn

Write in Sigma Notation, 4, 15, 26, …, 301

28

1

11 7n

n

04/19/23 20:21 63

Example 3

Find the following sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

11 1 21

2S

1

2nn a a

S

12164

Example 4

Find the following sum: 4, 15, 26, …, 301

427004/19/23 20:21 65

Your Turn

Find the following sum: 15, 11, 7, …, –61

46004/19/23 20:21 66

Example 5

Evaluate 32

1

2 8n

n

32

1

2 8n

n

1

2nn a a

S

1

/

n

k n

a

a

32 2 1 8

10 2 32 8 7204/19/23 20:21 67

Example 5

Evaluate

1312

32

1

2 8n

n

1

2nn a a

S

1

32

10

72n

n

a

a

32 10 72

2S

04/19/23 20:21 68

Evaluate

20

4

1

4 5n

n

04/19/23 20:21 69

Example 6

Evaluate

10,230

31

1

2 298n

n

04/19/23 20:21 70

Your Turn

Write the expression in expanded form and then find the sum.

= -5 + -3 + -1 + 1

= -8

4

1

72n

n

Write the expression in expanded form and then find the sum.

7

4

13b

b

= 80 + 242 + 728 + 2186

= 3236

Consider the Sequence

1,4,9,16,25,...a) Write down an expression for Sn.

2 2 2 2 21 2 3 4 ...nS n

2

1

n

k

k

Consider the Sequence

1,4,9,16,25,...b) Find Sn for n = 1, 2, 3, 4, and 5

1

2

3

4

5

1

1 4 5

1 4 9 14

1 4 9 16 30

1 4 9 16 25 55

S

S

S

S

S

2

1

n

k

k

Modeling Growth

• Is this an arithmetic series or geometric series?

• What is the common ration of the geometric series?

1111

0

08.1)08.01(08.0

4000

nnnnn uuuuu

u

Sum of a Finite Geometric Series

• The sum of the first n terms of a geometric series is

1(1 )

1

n

n

a rS

r

Notice – no last term needed!!!!

Formula for the Sum of a Finite Formula for the Sum of a Finite Geometric SeriesGeometric Series

r

raS

n

n 1

11

n = # of termsn = # of terms

aa1 1 = 1= 1stst term term

r = common ratior = common ratio

What is n? What is a1? What is r?

Example

• Find the sum of the 1st 10 terms of the geometric sequence: 2 ,-6, 18, -54

10 10

10

2(1 - (-3) ) 2(1 - 3 )S = =

1- -3 29,524

4

1(1 )

1

n

n

a rS

rWhat is n? What is a1? What is r?

That’s It!

Example: Consider the geometric Example: Consider the geometric series 4+2+1+½+… .series 4+2+1+½+… .

• Find the sum of the first 10 terms.

• Find n such that Sn=31/4.

r

raS

n

n 1

11

21

1

21

14

10

10S

128

1023

1024

20464

21

10241023

4

211024

11

410

S

21

1

21

14

4

31

n

21

1

21

14

4

31

n

2121

14

4

31

n

n

2

118

4

31

n

2

11

32

31n

2

1

32

1n

2

1

32

1

5n

n

n

2

1

32

1

n232 log232=n

Assignment

Arithmetic Arithmetic SeriesSeries

• When the famous mathematician C. F. Gauss was 7 years old, his teacher posed problem to the class and expected that it would keep the students busy for a long time.

– Gauss, though, answered it almost immediately.

• Suppose we want to find the sum of the numbers 1, 2, 3, 4, . . . , 100, that is,

• 1 + 2 + 3 + 4 + 5+ 6+ …+ 100

• His idea was this:

• Since we are adding numbers produced according to a fixed pattern, there must also be a pattern (or formula) for finding the sum.

– He started by writing the numbers from 1 to 100 and below them the same numbers in reverse order.

• Writing S for the sum and adding corresponding terms gives:

– It follows that 2S = 100(101) = 10,100 and so S = 5050.

1 2 3 98 99 100

100 99 98 3 2 1

2 101 101 101 101 101 101

S

S

S

• We want to find the sum of the first n terms of the arithmetic sequence whose terms are un = a1 + (n – 1)d.

– That is, we want to find:

11

1 1 1 1

1

1

2 3

1

n

nk

S a n d

a a d a d a d

a n d

• Using Gauss’s method, we write:

– There are n identical terms on the right side of this equation.

1 1 1 1

1 1 1 1

1 1 1 1

2 1

1 2

2 2 1 2 1 2 1 2 1

n

n

n

S a a d a n d a n d

S a n d a n d a d a

S a n d a n d a n d a n d

1 1 1

1

2 2 ( 1) ( 1)

2 ( 1)2

n

n

S n a n d a a n d

nS a n d

1

2nn a

Sa

1 1Substitute na a n d

1 1

1

1

2

2 1

2

n a a n dS

n a n dS

1

# of Terms

1st Term

Difference

n

a

d

• Find the sum of the first 40 terms of the arithmetic sequence

3, 7, 11, 15, . . .

– Here, a = 3 and d = 4.– Using Formula 1 for the partial sum of

an arithmetic sequence, we get:–

S40 = (40/2) [2(3) + (40 – 1)4] = 20(6 + 156) = 3240

Write the first three terms and the

Write the first three terms and the last two terms of the following

last two terms of the following arithmetic series.arithmetic series.

50

1

73 2p

p

71 69 67 . . . 25 27

What is the sum of What is the sum of

this series?this series?

71 69 67 . . . 25 27

27 25 . . . 67 69 71

44 44 44 . . . 44 44 44

50 71 27

2

110071 + (-27) Each sum is the same.

50 Terms

Find the sum of the terms of this arithmetic series.

35

1

29 3k

k

1

2nn a a

S

1

35

35

26

76

n

a

a

35 26 76

2875

S

S

Find the sum of the terms of this arithmetic series. 151 147 143 139 . . . 5

1

2nn a a

S

1

40

40

151

5

n

a

a

40 151 5

22920

S

S

1 1

5 151 1 4

40

na a n d

n

n

What term is -5?What term is -5?

35

1

45 5i

i

1

2nn a a

S

12 1

2

n a n dS

135 40 130nn a a 135 40 5n a d

35 40 130

21575

S

S

35 2 40 35 1 3

21575

S

S

Homework 6F Page 169 (1 –

11)Chose 9

For #8, Just do a) & b).

Formula for the Sum of a Finite Formula for the Sum of a Finite Geometric SeriesGeometric Series

r

raS

n

n 1

11

n = # of termsn = # of terms

aa1 1 = 1= 1stst term term

r = common ratior = common ratio

Example: Consider the geometric Example: Consider the geometric series 4+2+1+½+… .series 4+2+1+½+… .

• Find the sum of the first 10 terms.

• Find n such that Sn=31/4.

r

raS

n

n 1

11

21

1

21

14

10

10S

128

1023

1024

20464

21

10241023

4

211024

11

410

S

21

1

21

14

4

31

n

21

1

21

14

4

31

n

2121

14

4

31

n

n

2

118

4

31

n

2

11

32

31n

2

1

32

1n

2

1

32

1

5n

n

n

2

1

32

1

n232 log232=n

Homework

6G.1Page 171

(1 – 5)#2a (Conjugate denominator)

Infinite Geometric Series

• Consider the infinite geometric sequence

•

• What happens to each term in the series?

• They get smaller and smaller, but how small does a term actually get?

1 1 1 1 1, , , ,... ...

2 4 8 16 2

n

Each term approaches 0

Partial Sums

• Look at the sequence of partial sums:

1

2

3

121 1 32 4 41 1 1 72 4 8 8

S

S

S

What is happening to the

sum?

It is approaching 1

0

1

It’s CONVERGING

TO 1.

Here’s the Rule

Sum of an Infinite Geometric Sum of an Infinite Geometric SeriesSeries

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

aa11 + a + a11r + ar + a11rr22 + … + a + … + a11rrn n + …+ …

converges to the sumconverges to the sum

If If |r| > 1|r| > 1, then the series diverges , then the series diverges (does not have a sum)(does not have a sum)

1

1

aS

r

Converging – Has a Sum

• So, if -1 < r < 1, then the series will converge. Look at the series given by

• Since r = , we know that the sum

is

• The graph confirms:

1 1 1 1

...4 16 64 256

1

4

1

114

11 31

4

aS

r

Diverging – Has NO Sum

• If r > 1, the series will diverge. Look at 1 + 2 + 4 + 8 + ….

• Since r = 2, we know that the series grows without bound and has no sum.

• The graph confirms:

1

2

3

1

1 2 3

1 2 4 7...

S

S

S

Example

• Find the sum of the infinite geometric series 9 – 6 + 4 - …

• We know: a1 = 9 and r = ?

2

3

1 9 2721 51

3

aS

r

You Try

• Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + …

• Since r = -½ 1

124 24 48

161 3 31

2 2

aS

r

S

Page 173 6G.2

1

3 3 31. 0.3 ....

10 100 1000

.) Find i. ii. a u r

1

3.) i.

10a u

31100

.) ii. r 0.103 10

10

a

Page 173 6G.23 3 3

1. 0.3 ....10 100 1000

1.) Using , show that 0.a 3

3b

3 3 3.) 0.3 ...

10 100 1000

3 3 30.3 ...

10 100 1000n

b

S

1

310, then

10

1 1

1u

n Sr

10.3

3

Page 173 All 2 - Page 173 All 2 - 88

HomeworkHomework

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