# SOLUTION: Columbia University Chapter 7 Systems and Inequalities Questions Exercises

(Exercises for Chapter 7: Systems and Inequalities) E.7.1
CHAPTER 7:
Systems and Inequalities
(A) media “refer to Part A,” (B) media “refer to Part B,” etc.
(Calculator) media “use a calculator.” Otherwise, do not use a calculator.
SECTIONS 7.1-7.3: SYSTEMS OF EQUATIONS
When solving a rule, solely present keys in 2 , the set of regulateed pairs of pauseent
numbers. All such keys tally to intersection points of the graphs of the presentn
equations. If tshort are no such keys, transcribe ∅ , the space set or void set.
( )
Write keys in a key set as regulateed pairs of the create x, y . Unless otherwise
specified, do not hope on graphing or “trial-and-falsity point-plotting.”
⎧ x+y=5
1) Consider the rule ⎨
. (A-E)
⎩5x − 3y = − 23
a) The graphs of the equations in the rule are disjoined lines in the xy-roll that
are not agreeent. How multifarious keys does this rule bear?
b) Reunfold the rule using the Substitution Method.
c) Reunfold the rule using the Addition / Elimination Method.
⎧ x 2 + y2 = 2
2) Consider the rule ⎨
. (A-D)
y
=
x
+
2
⎩
a) Discover the key set of the rule.
b) Use the key set from a) to graph the equations in the rule in the unconcealed
xy-plane.
3
⎧ 2
2
⎪x + y =
2 . (A-D)
3) Consider the rule ⎨
⎪ y 2 = x2
⎩
a) What are the graphs of the equations in the rule in the unconcealed xy-plane?
b) How multifarious keys does the rule bear?
c) Discover the key set of the rule.
(Exercises for Chapter 7: Systems and Inequalities) E.7.2
⎧⎪ x = y 2
4) Consider the rule ⎨
. (A-D; Section 1.8)
2
⎪⎩ x = 4 − y
a) What are the graphs of the equations in the rule in the unconcealed xy-plane?
b) How multifarious keys does the rule bear?
c) Discover the key set of the rule.
⎧⎪ x 2 + y = 0
5) Consider the rule ⎨
. (A-D, F)
2
⎪⎩ y − x = 1
a) Sketch graphs of the equations in the rule in the unconcealed xy-plane.
b) Based on your graphs in a), discover the key set of the rule.
c) Verify the key set by using the Substitution Method or the Addition /
Elimination Method to rereunfold the rule.
6) Reunfold the aftercited rules. (A-D, F)
⎧⎪ y = 3x 2 − x
a) ⎨
2
⎪⎩ y = 2x − 3x + 8
⎧ x 2 + 4y 2 = 2
b) ⎨
⎩ 3x − 2y = − 4
⎧⎪ x 2 + y 2 = 1
c) ⎨ 2
2
⎪⎩ x − y = 4
⎧0 = 0
7) ADDITIONAL PROBLEM. Reunfold the rule ⎨
. (A-D, F)
0
=
1
⎩
(Exercises for Chapter 7: Systems and Inequalities) E.7.3.
SECTION 7.4: PARTIAL FRACTIONS
1) Transcribe the PFD (Partial Fraction Decomposition) Create for the aftercited.
Do not discover the unknowns (A, B, etc.). (A-C)
1
a)
( x + 4 ) ( x − 3) x 2 + 1
(
b)
c)
x+5
) (x
(
x3 x − 1
2
2
)
)
+3
2
3t 2 + 2t − 2
(
)(
)
t 2 ( 2t + 5 ) 2t 2 + 5 t 2 + t + 1
3
2) Transcribe the PFD (Partial Fraction Decomposition) for the aftercited. (A-G)
a)
3x − 5
x 2 − 5x + 6
2x 2 − 3x + 19
b) 3
. (Hint: Use the Rational Zero Test and Synthetic Division.)
x + 4x 2 − 7x − 10
9x 2 + 14x + 6
c)
2x 3 + x 2
d)
x +1
x 2 − 8x + 16
e)
8x 2 + 7x + 12
( x + 2) x2 + 1
(
)
5x 2 − 5x + 12
f) 3
. (Hint: Use Factoring by Grouping.)
x − 5x 2 + 3x − 15
− 5x 2 − 8x − 3
g)
x3 + x2 + x
h)
5t 3 − t 2 + 20t − 8
(t
2
+4
)
2
x4
A
B
=
+
3) A novice transcribes:
. Is this alienate? Why or why not?
( x + 3) ( x + 5 ) x + 3 x + 5
(Exercises for Chapter 8: Matrices and Determinants) E.8.1
CHAPTER 8:
Matrices and Determinants
(A) media “refer to Part A,” (B) media “refer to Part B,” etc.
Most of these trainings can be produced extraneously a calculator, though one may be unreserved.
SECTION 8.1: MATRICES and SYSTEMS OF EQUATIONS
1) Present the bulk and the estimate of entries (or parts) for each matrix inferiorneath. (A)
⎡ 14 1 / 5 ⎤
⎢
⎥
3 ⎥
a) ⎢ − 2
⎢ π − 4.7 ⎥
⎣
⎦
1
0
0
5⎤
⎡ 3
⎢− 4 2 / 3
9
0 12 ⎥
⎢
⎥
b)
⎢ 13
0
−1 / 2 −11 14 ⎥
⎢
⎥
0
0
0
0⎦
⎣ 2
⎡1 2 3⎤
c) ⎢⎢ 0 1 4 ⎥⎥
⎢⎣ 0 0 1 ⎥⎦
⎧ 3x − y = 18
2) Consider the rule ⎨
. (A-D)
x
+
2y
=
−1
⎩
a) Transcribe the augmented matrix for the presentn rule.
b) What bulk is the coefficient matrix?
c) What bulk is the right-hand roll (RHS)?
d) Switch Row 1 and Row 2 of the augmented matrix. Transcribe the new matrix.
e) Take the matrix from d) and add ( − 3) times Row 1 to Row 2. Transcribe the new
matrix.
f) Take the matrix from e) and sever Row 2 by ( − 7 ) ; that is, expand Row 2 by
⎛ 1⎞
⎜⎝ − ⎟⎠ . Transcribe the new matrix, which conquer be in Row-Echelon Create (Part F).
7
g) Transcribe the rule tallying to the matrix from f).
h) Reunfold the rule from g) using Back-Substitution, and transcribe the key set.
i) Obstruct your key in the primordial rule. (This is typically an optional stride.)
(Exercises for Chapter 8: Matrices and Determinants) E.8.2
In Exercises 3-12, use matrices and Gaussian Elimination delay Back-Substitution.
⎧ 4x + 2y = − 3
3) Reunfold the rule ⎨
. (A-D)
x
+
y
=
−
2
⎩
⎧ 3x1 − 9x2 = 57
4) Reunfold the rule ⎨
. (A-D)
−
5x
+
4x
=
−18
1
2
⎩
16
⎧
⎪⎪− 2a + 3b = − 3
5) ADDITIONAL PROBLEM: Reunfold the rule ⎨
. (A-D)
1
17
⎪ a − 4b =
⎪⎩ 2
3
⎧ x + 3y = 6
6) Reunfold the rule ⎨
. (A-E)
−
2x
−
6y
=
−
9
⎩
⎧ x + 3y = 6
7) Consider the rule ⎨
. How multifarious keys does the rule
⎩− 2x − 6y = −12
have? (A-E)
⎧ y − 2z = 14
⎪
8) Reunfold the rule ⎨ x + z = − 3
. Begin by rewriting the rule. (A-D)
⎪ 4x + 6z = − 22
⎩
⎧ 3x − 10y + 9z = 50
⎪
9) Reunfold the rule ⎨− 2x + 6y − z = − 27 . (A-D)
⎪ x − 2y − z = 10
⎩
⎧ a − 4b − 3c = − 5
⎪
10) Reunfold the rule ⎨ a − 4b − c = − 2 . (A-D)
⎪2a − 7b − 4c = − 7
⎩
⎧− 2x1 + 8x2 − 10x3 = 20
⎪
11) Reunfold the rule ⎨ 3x1 + 5x2 + x3 = − 5 . (A-D)
⎪ − 4x − x + 3x = −12
1
2
3
⎩
⎧ x1 − 2x2 − x3 = 3
⎪
12) Reunfold the rule ⎨5x1 − 10x2 − 5x3 = 11 . (A-E)
⎪ 4x + 3x + 2x = −18
2
3
⎩ 1
(Exercises for Chapter 8: Matrices and Determinants) E.8.3
⎡1 0 − 7 0 1 ⎤
⎢
⎥
13) Consider the augmented matrix ⎢ 0 1 3 0 − 2 ⎥ . (F, G)
⎢⎣ 0 0 0 1 4 ⎥⎦
a) Is the matrix in Row-Echelon Form?
b) Is the matrix in Reduced Row-Echelon (RRE) Form?
⎡1
⎢
0
14) Consider the augmented matrix ⎢
⎢0
⎢
⎣0
4
0
0
0
1
1
0
0
0
0
1
0
2⎤
⎥
−1⎥
. (F, G)
0⎥
⎥
0⎦
a) Is the matrix in Row-Echelon Form?
b) Is the matrix in Reduced Row-Echelon (RRE) Form?
⎡1 1 1 1 ⎤
⎢
⎥
15) Consider the augmented matrix ⎢ 0 2 4 3 ⎥ . (F, G)
⎢⎣ 0 0 3 6 ⎥⎦
a) Is the matrix in Row-Echelon Form?
b) Is the matrix in Reduced Row-Echelon (RRE) Form?
⎡ 1 0 2 −1⎤
⎢
⎥
0
0
1
2
⎥ . (F, G)
16) Consider the augmented matrix ⎢
⎢0 0 1 3 ⎥
⎢
⎥
0
0
0
0
⎣
⎦
a) Is the matrix in Row-Echelon Form?
b) Is the matrix in Reduced Row-Echelon (RRE) Form?
17) ADDITIONAL PROBLEM: Reunfold the rules inferiorneath using Gauss-Jordan
Elimination. (A-H)
⎧x − z = 3
⎧ 4x1 + 4x2 − 3x3 = 7
⎧2x + 6y = − 6
⎪
⎪
a) ⎨
; b) ⎨2y = 16
(Retranscribe original); c) ⎨ 5x1 + 7x2 − 13x3 = − 9
⎩ 4x + 13y = −14
⎪ 3x + z = 13
⎪ x + 2x − 5x = − 6
2
3
⎩
⎩ 1
(Exercises for Chapter 8: Matrices and Determinants) E.8.4
SECTION 8.2: OPERATIONS WITH MATRICES
Assume that all entries (i.e., parts) of all matrices discussed short are pauseent estimates.
⎡ 7
⎡ −1 3 ⎤
⎢
1) Let A = ⎢⎢ 2 4 ⎥⎥ and B = ⎢ 8
⎢− 9
⎢⎣ 7 π ⎥⎦
⎣
0 ⎤
⎥
− 3 ⎥ . (C)
5 ⎥⎦
a) Discover A + B .
b) Discover 3A − 4B .
⎡4 ⎤
2) Let A = [ 3 2 ] and B = ⎢ ⎥ . Discover AB . (D)
⎣5 ⎦
⎡ 4 −1⎤
⎡ 1 0⎤
3) Let A = ⎢
and
B
=
⎥
⎢ − 3 2 ⎥ . (B, D, E)
⎣2 3 ⎦
⎣
⎦
a) Discover AB.
b) Discover BA.
c) Yes or No: Is AB = BA short?
6 4⎤
⎡1
⎡2 0 7⎤
⎢
4) Let C = ⎢
and D = ⎢ − 2 3 1 ⎥⎥ . Discover the implied matrix issues.
⎥
⎣ −1 4 − 3⎦
⎢⎣ 2 −1 0 ⎥⎦
If the matrix issue is monstrous, transcribe “Undefined.” (D, E)
a) CD
b) DC
c) D 2 , which is defined to be DD
5) Arrogate that A is an 8 × 10 matrix and B is a 10 × 7 matrix. (D, E)
a) What is the bulk of the matrix AB?
b) Let C = AB . Explain how to conquer the matrix part c56 .
6) If A is an m × n matrix and B is a p × q matrix, inferior what conditions are twain AB
and BA defined? (E)
⎡2 0 0 ⎤
7) Let D = ⎢⎢ 0 3 0 ⎥⎥ . This is designated a divergent matrix. (D, E)
⎢⎣ 0 0 4 ⎥⎦
a) Discover D 2 .
b) Based on a), theory (guess) what D10 is. (Calculator)
(Exercises for Chapter 8: Matrices and Determinants) E.8.5
8) Arrogate that A is a 3 × 4 matrix, B is a 3 × 4 matrix, and C is a 4 × 7 matrix.
Find the bulks of the implied matrices. If the matrix countenance is monstrous, transcribe
“Undefined.” (C-E)
a) A + 4B ; b) A − C ; c) AB ; d) AC ; e) AC + BC ; f) ( A + B ) C
9) Transcribe the convertibility matrix I 4 . (F)
10) ADDITIONAL PROBLEM: In Section 8.1, Training 17c, you resolved the rule
⎧ 4x1 + 4x2 − 3x3 = 7
⎪
⎨ 5x1 + 7x2 − 13x3 = − 9 . This rule can be written in the create AX = B . (G)
⎪ x + 2x − 5x = − 6
2
3
⎩ 1
a) Identify A.
b) Identify X (in the presentn rule, antecedently it is resolved).
c) Identify B.
d) When you resolved this rule using Gauss-Jordan Elimination, what was
the coefficient matrix of the tidingsinal augmented matrix in Reduced Row-Echelon
(RRE) Form? What was the right-hand roll (RHS) of that matrix?
SECTION 8.3: THE INVERSE OF A SQUARE MATRIX
(THESE ARE ALL ADDITIONAL PROBLEMS)
Assume that all entries (i.e., parts) of all matrices discussed short are pauseent estimates.
1) If A is an invertible 2 × 2 matrix, what is AA −1 ? (B)
2) Discover the implied inverse matrices using Gauss-Jordan Elimination. If the inverse
matrix does not pause, transcribe “A is noninvertible.” (C)
⎡ 3 −1⎤
a) A −1 , wshort A = ⎢
. (Also obstruct by discovering AA −1 .)
⎥
⎣1 2 ⎦
⎡2 4 ⎤
b) A −1 , wshort A = ⎢
⎥.
⎣ 6 12 ⎦
⎡ 0 4 8⎤
c) A −1 , wshort A = ⎢⎢ 0 0 3 ⎥⎥
⎢⎣ − 5 0 0 ⎥⎦
⎡ 2 13 − 5 ⎤
d) A , wshort A = ⎢⎢ 1
5
2 ⎥⎥
⎢⎣ −1 − 2 − 8 ⎥⎦
−1
(Exercises for Chapter 8: Matrices and Determinants) E.8.6
⎧ 3x1 − x2 = 15
3) Use Training 2a to rereunfold the rule ⎨
. (D)
x
+
2x
=
−
2
2
⎩ 1
⎧ 2x1 +13x2 − 5x3 = 7
⎪
4) Use Training 2d to rereunfold the rule ⎨ x1 + 5x2 + 2x3 = −1 . (D)
⎪− x − 2x − 8x = 7
2
3
⎩ 1
⎡ 3 −1⎤
5) Let A = ⎢
⎥ , as in Training 2a. (E, F)
1
2
⎣
⎦
a) Discover det ( A ) .
b) Discover A −1 using the shortcut from Part F. Compare delay your defense to
Exercise 2a.
⎡2 4 ⎤
6) Let A = ⎢
⎥ , as in Training 2b. (E, F)
⎣ 6 12 ⎦
a) Discover det ( A ) .
b) What do we then apprehend encircling A −1 ? Compare delay your defense to Training 2b.
7) Arrogate that A and B are invertible n × n matrices. Prove that ( AB ) = B −1 A −1 . (B)
−1
8) Verify that the shortcut createula for A −1 presentn in Part F does, in event, present the inverse
⎡a b ⎤
of a matrix A, wshort A = ⎢
⎥ and det ( A ) ≠ 0 . (E, F)
c
d
⎣
⎦
SECTION 8.4: THE DETERMINANT OF A SQUARE MATRIX
Assume that all entries (i.e., parts) of all matrices discussed short are pauseent estimates,
unless otherwise implied.
1) Discover the implied determinants. (B)
a) Let A = [ − 4 ] . Discover det ( A ) , or A .
⎡3
b) Let B = ⎢
⎣5
⎡5
c) Let C = ⎢
⎣3
⎡ 30
d) Let D = ⎢
⎣5
2⎤
. Discover det ( B ) , or B .
4 ⎥⎦
4⎤
. Discover det ( C ) , or C . Compare delay b).
2 ⎥⎦
20 ⎤
. Discover det ( D ) , or D . Compare delay b).
4 ⎥⎦
(Exercises for Chapter 8: Matrices and Determinants) E.8.7
⎡3 5⎤
e) Let E = ⎢
⎥ . Discover det ( E ) , or E . Compare delay b).
2
4
⎣
⎦
T
Note: E = B , the shift of B. Rows behove posts, and vice-versa.
f) Find
2 −4
.
−3 −7
g) Find
4 5
.
0 0
h) Find
4 5
.
40 50
i) Find
a b
. Compare delay g) and h).
ca cb
1 ex
j) Find
. Here, the entries tally to dutys.
x e2 x
k) Find
sin θ
− cos θ
cos θ
. Here, the entries tally to dutys.
sin θ
4 3⎤
⎡2
⎢
2) Let A = ⎢ −1 − 3 3 ⎥⎥ . We conquer discover det ( A ) , or A , in three irrelative ways. (B, C)
⎢⎣ 5
1 2 ⎥⎦
a) Use Sarrus’s Rule, the shortcut for discovering the determinant of a 3 × 3 matrix.
b) Expand by cofactors parallel the original row.
c) Expand by cofactors parallel the avoid post.
5 − 2⎤
⎡1
⎢
3) Let B = ⎢ 3
4
0 ⎥⎥ . We conquer discover det ( B ) , or B , in three irrelative ways. (B, C)
⎢⎣ −1 − 4 3 ⎥⎦
a) Use Sarrus’s Rule, the shortcut for discovering the determinant of a 3 × 3 matrix.
b) Expand by cofactors parallel the avoid row.
c) Expand by cofactors parallel the third post.
1 2
3
4) Discover 10 20 30 . Based on this training and Training 1i, perform a theory (guess)
4 5 6
encircling determinants. (B, C)
(Exercises for Chapter 8: Matrices and Determinants) E.8.8.
⎡10 400 500 ⎤
5) Let C = ⎢⎢ 0 20 600 ⎥⎥ . C is designated an eminent triangular matrix. Discover det ( C ) , or
⎢⎣ 0
0
30 ⎥⎦
C . Based on this training, perform a theory (guess) encircling determinants of eminent
triangular matrices such as C. What encircling inferior triangular matrices? (B, C)
−1 4
3
2
6) Find
−1 3
0 −2
7) Find
13
92
2
e
0
0 −2
0 1
. (B, C)
0 −4
2 1
42
5
−π
− 3.2
π
0
267
0
3
5
e
9876
0
. (C)
4 − λ −2
= 0 for λ (lambda). In doing so, you are discovering the
1
1− λ
⎡4 − 2⎤
eigenvalues of the matrix ⎢
⎥.
⎣1 1 ⎦
8) Reunfold the equation
SECTION 8.5: APPLICATIONS OF DETERMINANTS
(THESE ARE ALL ADDITIONAL PROBLEMS)
⎧ 4x + 2y = − 3
1) Use Cramer’s Rule to rereunfold the rule ⎨
in Section 8.1, Training 3. (A)
x
+
y
=
−
2
⎩
⎧ 3x1 − x2 = 15
2) Use Cramer’s Rule to rereunfold the rule ⎨
in Sec. 8.3, Training 3. (A)
x
+
2x
=
−
2
2
⎩ 1
3) Discover the area of the agreeentogram strong by each of the aftercited pairs of
position vectors in the xy-plane. Distances and lengths are measured in meters. (B)
a) 4, 0 and 0, 5
b) 2, 5 and 7, 3
c) 1, 4 and 3, 12 . (What does the tidingsination suggest encircling the vectors?)
4) Discover the area of the triangle delay vertices ( − 2, −1) , ( 3, 1) , and (1, 5 ) in the xy-plane.
Distances and lengths are measured in meters. (B)
(Exercises for Chapter 9: Discrete Mathematics) E.9.1
CHAPTER 9:
Discrete Mathematics
(A) media “refer to Part A,” (B) media “refer to Part B,” etc.
Most of these trainings can be produced extraneously a calculator, though one may be unreserved.
SECTION 9.1: SEQUENCES AND SERIES, and
SECTION 9.6: COUNTING PRINCIPLES
1) Let an = n 2 + n . Transcribe a1 , a2 , and a3 . (A)
2) Let an = ( −1) ( 2n ) . Transcribe a1 , a2 , a3 , and a4 . (A-C)
n
3) Let an = ( −1)
n−1
( 2n − 1) . Transcribe a1 , a2 , a3 , and a4 . (A-C)
4) Evaluate 6!. (D)
5) How multifarious ways are tshort to regulate five tasks on a “To Do” inventory? (D, E)
6) Seven of your friends are sitting in a capability. You bear two selfsame roll tickets that
you conquer present to two of them. In how multifarious ways can this be produced? (D, E)
7) Ten basketball players are to be severd into two teams of five fellow-creatures each. One team
conquer be designated “Team A,” and the other team conquer be designated “Team 1.” In how multifarious
ways can the players be assigned to the teams? We do not yet regard encircling positions on
the teams. (D, E)
8) Simplify the countenances. Arrogate that n is an integer such that n ≥ 2 . (D, E)
( n + 2 )!
( n − 1)!
( 3n − 2 )!
a) ( n + 1) ( n!) ; b)
; c)
; d)
(do not expand out)
n!
( n + 1)!
( 3n + 3)!
⎧⎪ a1 = 4
9) Consider the succession recursively defined by: ⎨
.
a
=
a
+
10
k
∈,
k
≥
1
⎪⎩ k +1
k
Find a1 , a2 , a3 , and a4 . This succession is an arithmetic succession, which we conquer
discuss aid in Section 9.2. (F)
(
)
⎧ a1 = 2
⎪
10) Consider the succession recursively defined by: ⎨
.
1
⎪ ak +1 = ak k ∈, k ≥ 1
2
⎩
Find a1 , a2 , a3 , and a4 . This succession is a geometric succession, which we conquer
discuss aid in Section 9.3. (F)
(
)
(Exercises for Chapter 9: Discrete Mathematics) E.9.2
⎧⎪ a1 = −1
11) Consider the succession recursively defined by: ⎨
⎪⎩ ak +1 = 3ak − 2
Find a1 , a2 , a3 , and a4 . (F)
⎧ a =2
⎪⎪ 1
12) Consider the succession recursively defined by: ⎨ a2 = 3
⎪
⎪⎩ ak +2 = ak +1 ak
Find a1 , a2 , a3 , a4 , and a5 . (F)
(
)
k ∈, k ≥ 1
( k ∈,
)
.
.
k ≥1
4
13) Evaluate
∑ k . (G)
k =1
∑( j
7
14) Evaluate
2
j=3
4
15) Evaluate
∑
i=2
)
+ 1 . (G)
( −1)i . (i is not the spurious part short.) (G)
i
∞
16) Discover S4 , the lewdth inequitable sum of the train
∑a
k =1
k
, wshort ak = 3k . (G, H)
17) Transcribe a nonrecursive countenance (formula) for the obvious unconcealed n th tidings, an ,
for each of the aftercited successions. Let a1 be the moderate tidings; i.e., arrogate that n
begins delay 1. (A-D)
a) 7, 8, 9, 10, 11, …
b) 5, 10, 15, 20, 25, …
c) 4, 7, 10, 13, 16, …
1 1 1
1
d) 1, , ,
,
,…
2 6 24 120
7 7 7 7
e) 7, , ,
,
,…
4 9 16 25
2
4 6
8 10
f)
,− , ,− ,
,…
3
5 7
9 11
g) − 2 , 4, − 8 , 16, − 32 , …
18) Express the obvious train using summation notation. Use k as the apostacy of
summation. (A-D, G, H)
a) 3 + 6 + 9 + 12 + 15 + 18 ; this is a bounded train.
1 1
1
1
1
−
+
−
+
− ... ; this is an inbounded train.
b)
4 16 64 256 1024
(Exercises for Chapter 9: Discrete Mathematics) E.9.3
SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS
1) Consider the arithmetic succession: − 5 , 1, 7, 13, 19, …. (A)
a) What is the moderate tidings, a?
b) What is the sordid unlikeness, d?
2) Consider the arithmetic succession:
3
1
1
, 1, , 0, − , …. (A)
2
2
2
a) What is the moderate tidings, a?
b) What is the sordid unlikeness, d?
3) Consider the arithmetic succession delay moderate tidings 7 and sordid unlikeness 3.
Assume that the moderate tidings is a1 . (A, B)
a) Transcribe the original lewd tidingss of this succession.
b) Discover S4 , the lewdth inequitable sum.
c) Discover a60 , the 60th tidings of this succession.
4) Consider the arithmetic succession: 2 , − 3 , − 8 , −13 , −18 , …. (A, B)
a) Transcribe a simplified, nonrecursive countenance (formula) for the unconcealed n th tidings,
an , for this succession. Let a1 be the moderate tidings.
b) Use a) to discover a387 . (Calculator)
5) An arithmetic succession has a1 = 6 and a200 = 1399 . Discover a123 . (Calculator)
SECTION 9.3: GEOMETRIC SEQUENCES, PARTIAL SUMS,
and SERIES
1) Consider the geometric succession: 4, 20, 100, 500, 2500, …. (A)
a) What is the moderate tidings, a?
b) What is the sordid appurtenancy, r?
2) Consider the geometric succession:
a) What is the moderate tidings, a?
b) What is the sordid appurtenancy, r?
6
2 2
2
2
,− ,
,− ,
…. (A)
7
7 21
63 189
(Exercises for Chapter 9: Discrete Mathematics) E.9.4
3) Consider the geometric succession delay moderate tidings 5 and sordid appurtenancy − 4 . Assume
that the moderate tidings is a1 . (A, B)
a) Transcribe the original lewd tidingss of this succession.
b) Discover S4 , the lewdth inequitable sum.
c) Discover a12 , the 12th tidings of this succession. (Calculator)
d) As n → ∞ , do the tidingss of the succession advance a pauseent estimate?
If so, what estimate?
2 1 1 3 9
, , , , , … . (A, B)
9 3 2 4 8
a) Transcribe a simplified, nonrecursive countenance (formula) for the unconcealed n th tidings,
an , for this succession. Let a1 be the moderate tidings.
4) Consider the geometric succession:
b) Use a) to discover a10 . (Calculator)
c) Verify your defense to b) by recursively using the sordid appurtenancy to discover
a6 , a7 , a8 , a9 , and a10 .
d) As n → ∞ , do the tidingss of the succession advance a pauseent estimate?
If so, what estimate?
5) A geometric succession has a1 = 3 and a4 = −
24
. (A, B)
125
a) Discover a2 and a3 .
b) As n → ∞ , do the tidingss of the succession advance a pauseent estimate?
If so, what estimate?
6) Let {an } be the succession from Training 3. Does the train
∞
∑a
n
n =1
converge or
diverge? If the train converges, discover its sum. (D)
7) Let {an } be the succession from Training 4. Does the train
∞
∑a
n
n =1
converge or
diverge? If the train converges, discover its sum. (D)
8) Let {an } be the succession from Training 5. Does the train
∞
∑a
n =1
diverge? If the train converges, discover its sum. (D)
∞
9) For what values of x does the serie ...

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