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Copyright © 2014, 2010 Pearson Education, Inc.
Chapter 9
Further Topics in Algebra
Copyright © 2014, 2010 Pearson Education, Inc.
Copyright © 2014, 2010 Pearson Education, Inc.
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
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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.
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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
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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
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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
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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.
Example: Finding a General Term of a Sequence from a Pattern
: 1 2 3 4 5 . . .
term: 1, 4, 9, 16, 25, . . . , an
n n
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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.
Example: Finding a General Term of a Sequence from a Pattern
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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
Example: Finding a General Term of a Sequence from a Pattern
11 .
n
n
na
n
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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.
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Simplify.
= 16 · 15 = 240
= (n + 1)n
Example: Simplifying a Factorial Expression
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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
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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
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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
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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
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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
Example: Evaluating Sums Given in Summation Notation
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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.
Example: Evaluating Sums Given in Summation Notation
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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
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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
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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
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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
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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
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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
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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
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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).
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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.
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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.
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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.
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PASCAL’S TRIANGLE
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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
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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
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DEFINITION OF
If r and n are integers with 0 ≤ r ≤ n, then we define
n
r
n!
r! n r !
n
r
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Evaluate each binomial coefficient.
n
r
Example: Evaluating
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Example: Evaluating n
r
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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
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Find the binomial expansion of (x – 3y)4.
(x – 3y)4 = [x + (–3y)]4
Example: Expanding a Binomial Power by Using the Binomial Theorem
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Example: Expanding a Binomial Power by Using the Binomial Theorem
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Example: Expanding a Binomial Power by Using the Binomial Theorem
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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 .
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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
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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