Copyright © 2014, 2010 Pearson Education, Inc. Chapter 9 Further Topics in Algebra Copyright ©...

Copyright © 2014, 2010 Pearson Education, Inc.

Chapter 9

Further Topics in Algebra



Section 9.1 Sequences ands Series

1. Use sequence notation and find specific and general terms in a sequence.

2. Use factorial notation.3. Use summation notation to write partial sums of a series.

SECTION 1.1

Copyright © 2014, 2010 Pearson Education, Inc. 3

DEFINITION OF A SEQUENCE

An infinite sequence is a function whose domain is the set of positive integers. The function values, written as

a1, a2, a3, a4, … , an, …,

are called the terms of the sequence. The nth term, an, is called the general term of the sequence.


Write the first six terms of the sequence defined by:

1 11n

nb n

1 1 2

1

11 1 1 1

1b

2 1 3

2

1 1 11 1

2 2 2b

Replace n with each integer from 1 to 6.

Example: Writing the Terms of a Sequence from the General Term


3 1 4

3

1 1 11 1

3 3 3b

4 1 5

4

1 1 11 1

4 4 4b

6 1 7

6

1 1 11 1

6 6 6b

5 1 6

5

1 1 11 1

5 5 5b

Example: Writing the Terms of a Sequence from the General Term


Write the general term an for a sequence whose first five terms are given.

Write the position number of the term above each term of the sequence and look for a pattern that connects the term to the position number of the term.

Example: Finding a General Term of a Sequence from a Pattern

1 2 3 4a. 1, 4, 9, 16, 25, ... b. 0, , , , ,...

2 3 4 5


a.

Apparent pattern: Here 1 = 12, 4 = 22, 9 = 32, 16 = 42, and 25 =52. Each term is the square of the position number of that term. This suggests an = n2.


: 1 2 3 4 5 . . .

term: 1, 4, 9, 16, 25, . . . , an

n n


b.

Apparent pattern: When the terms alternate in sign and n = 1, we use factors such as (−1)n if we want to begin with the factor −1 or we use factors such as (−1)n+1 if we want to begin with the factor 1.



Notice that each term can be written as a quotient with denominator equal to the position number and numerator equal to one less than the position number, suggesting the general term


11 .

n

n

na

n


DEFINITION OF FACTORIAL

For any positive integer n, n factorial (written n!) is defined as

As a special case, zero factorial (written 0!) is defined as

! 1 4 3 2 1.n n n

0! 1.


Simplify.

= 16 · 15 = 240

= (n + 1)n

Example: Simplifying a Factorial Expression


Write the first five terms of the sequence whose general term is:

Replace n with each integer from 1 through 5.

11

!

n

na n

1 1 2

1 1

1 11

! 1a

2 1 3

2 2

1 1 1

! 2 1 2a

Example: Writing Terms of a Sequence Involving Factorials


1

5

5 61 1 1

! 5 4 35 2 1 120a

3 1 4

3 3

1 1 1

! 3 2 1 6a

1 5

4

41 1 1

! 4 34 2 1 24a

Example: Writing Terms of a Sequence Involving Factorials


SUMMATION NOTATION

The sum of the first n terms of a sequence a1, a2, a3, …, an, … is denoted by

The letter i in the summation notation is called the index of summation, n is called the upper limit, and 1 is called the lower limit, of the summation.

1 2 31

.n

i ni

a a a a a


Find each sum.9

1

i

ia.

a. Replace i with integers 1 through 9, inclusive, and then add.

7

2

4

2 1j

j

b.4

0

2

!

k

k kc.

9

1

2 3 4 41 95 6 7 8 5i

i

Example: Evaluating Sums Given in Summation Notation


b. Replace j with integers 4 through 7, inclusive, and then add.

2 22

2

4

2

7

2 1 2 1 2 1

2 1 2 1

31 49 71 97

4

7

5

6

248

j

j



0 41 2 34

0 0 4

2 2 2 2 2 2

! ! ! ! ! !

1 2 4 8 16

1 1 2 6 244 2

1

3

23

1

2 73

2

k

k k

c. Replace k with integers 0 through 4, inclusive, and then add.



SUMMATION PROPERTIES

Let ak and bk, represent the general terms of two sequences, and let c represent any real number. Then

1. ck1

n

cn

2. cakk1

n

c akk1

n


SUMMATION PROPERTIES

3. ak bk k1

n

ak k1

n

bkk1

n

4. ak bk k1

n

ak k1

n

bkk1

n

5. akk1

n

ak k1

j

akkj1

n

, for 1 j n


DEFINITION OF A SERIES

Let a1, a2, a3, … , ak, … be an infinite sequence. Then

1. The sum of the first n terms of the sequence is called the nth partial sum of the sequence and is denoted by

This sum is a finite series.1 2 3

1

.n

n ii

a a a a a


DEFINITION OF A SERIES

2. The sum of all terms of the infinite sequence is called an infinite series and is denoted by

1 2 31

.n ii

a a a a a


Write each sum in summation notation.

3 5 7 21 a.

10

1

3 5 7 21 2 1k

k

a. This is the sum of consecutive odd integers from 3 to 21. Each can be expressed as 2k + 1, starting with k = 1 and ending with k = 10.

1 1 1

4 9 49 b.

Example: Writing a Partial Sum in Summation Notation


b. This finite series is the sum of fractions, each of which has numerator 1 and denominator k2, starting with k = 2 and ending with k = 7.

7

22

1 1 1 1

4 9 49 k k

Example: Writing a Partial Sum in Summation Notation


Section 9.5 The Binomial Theorem

1. Use Pascal’s Triangle to compute binomial coefficients.2. Use Pascal’s Triangle to expand a binomial power.3. Use the Binomial Theorem to expand a binomial power.4. Find the coefficient of a term in a binomial expansion.

SECTION 1.1


BINOMIAL EXPANSIONSThe patterns for expansions of (x+ y)n (with n = 1, 2, 3, 4, 5) suggest the following:

1. The expansion of (x+ y)n has n + 1 terms.

2. The sum of the exponents on x and y in each term equals n.

3. The exponent on x starts at n (xn = xn · y0) in the first term and decreases by 1 for each term until it is 0 in the last term (x0 · yn = yn).


BINOMIAL EXPANSIONS

4. The exponent on y starts at 0 in the first term (xn = xn · y0) and increases by 1 for each term until it is n in the last term (x0 · yn = yn).

5. The variables x and y have symmetrical roles. That is, replacing x with y and y with x in the expansion of (x+ y)n yields the same terms, just in a different order.


BINOMIAL EXPANSIONS

You may also have noticed that the coefficients of the first and last terms are both 1 and the coefficients of the second and the next-to-last terms are equal.

In general, the coefficients of xn−jyj and xjyn−j are equal for j = 0, 1, 2, …, n.

The coefficients in a binomial expansion of (x + y)n are called the binomial coefficients.


PASCAL’S TRIANGLEWhen expanding (x + y)n the coefficients of each term can be determined using Pascal’s Triangle.

The top row of the triangle, which contains only the number 1, represents the coefficients of (x + y)0 and is referred to as the zeroth row. The next row, called the first row, represents the coefficients of (x + y)1.

Each row begins and ends with 1. Each entry of Pascal’s Triangle is found by adding the two neighboring entries in the previous row.


PASCAL’S TRIANGLE


Expand (4y – 2x)5.

x y 5 x5 5x4y 10x3y2 10x2y3 5xy4 y5

The fifth row of Pascal’s Triangle yields the binomial coefficients 1, 5, 10, 10, 5, 1.

Replace x with 4y and y with –2x.

Example: Using Pascal’s Triangle to Expand a Binomial Power


554 2x y y x

Expanding a difference results in alternating signs.

5 4 3 2

2 3 4 5

1024 2560 2560

1280 320 32

y y x y x

y x yx x

5 4 3 2

2 3 4 5

4 5 4 2 10 4 2

10 4 2 5 4 2 2

y y x y x

y x y x x

Example: Using Pascal’s Triangle to Expand a Binomial Power


DEFINITION OF

If r and n are integers with 0 ≤ r ≤ n, then we define

n

r

n!

r! n r !

n

r


Evaluate each binomial coefficient.

n

r

Example: Evaluating


Example: Evaluating n

r


THE BINOMIAL THEOREMIf n is a natural number, then the binomial expansion of (x + y)n is given by

1 2 2

0 1 2n n n n

n r r n

n n nx y x x y x y

n nx y y

r n

0

.n

n r r

i

nx y

r

The coefficient of xn–ryr isn

r

!.

! !

n

r n r


Find the binomial expansion of (x – 3y)4.

(x – 3y)4 = [x + (–3y)]4

Example: Expanding a Binomial Power by Using the Binomial Theorem


PARTICULAR TERM IN A BINOMIAL EXPRESSION

The term containing the factor xr in the expansion of (x + y)n is

n

n r

xryn r .


Find the term containing x10 in the expansion of (x + 2a)15 .

Use the formula for the term containing xr.

15 1010152

15 10r n rnx y x a

n r

510152

5x a

Example: Finding a Particular Term in a Binomial Expansion


10 5 515!

25! 15 5 !

x a

10 5

10 5

10 5

15!32

5! 10 !

15 14 13 12 11 10!32

5! 10!

96,096

x a

x a

x a

Example: Finding a Particular Term in a Binomial Expansion

Copyright © 2014, 2010 Pearson Education, Inc. Chapter 9 Further Topics in Algebra Copyright ©...

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