 9.1.1: For the function find and f1x2 = f122 f132.
 9.1.2: True or False A function is a relation between two sets D and R so ...
 9.1.3: A(n) is a function whose domain is the set of positive integers.
 9.1.4: True or False The notation represents the fifth term of a sequence.
 9.1.5: If is an integer, then
 9.1.6: The sequence is an example of a sequence.
 9.1.7: The notation is an example of notation.
 9.1.8: True or False a n k=1 k = 1 + 2 + 3 + + n = n1n + 12 2
 9.1.9: 10!
 9.1.10: 9!
 9.1.11: 9! 6!
 9.1.12: 12! 10!
 9.1.13: 3! 7! 4!
 9.1.14: 5! 8! 3!
 9.1.15: {sn} = 5n6
 9.1.16: {sn} = 5n2 + 16
 9.1.17: {an} = e n n + 2 f
 9.1.18: {bn} = e 2n + 1 2n f
 9.1.19: {cn} = 5112n+1 n2 6
 9.1.20: {dn} = e112n1 a n 2n  1 bf
 9.1.21: {sn} = b 2n 3n + 1 r
 9.1.22: {sn} = e a 4 3 b n f
 9.1.23: {tn} = b 112n 1n + 121n + 22 r
 9.1.24: {an} = b 3n n r
 9.1.25: {bn} = b n en r
 9.1.26: {cn} = b n2 2n r
 9.1.27: 4 5 , 3 4 , 2 3 , 1
 9.1.28: 1 3 # 4 , 1 2 # 3 , 1
 9.1.29: 1, 1 8 ,
 9.1.30: 6 81 , 8 27 , 4 9 , 2
 9.1.31: 1, 1, 1, 1, 1, 1,
 9.1.32: 1, 3, 5, 7, 1 8 ,
 9.1.33: 1, 2, 3, 4, 5, 6,
 9.1.34: 2, 4, 6, 8, 10,
 9.1.35: a1 = 2; an = 3 + an1
 9.1.36: a1 = 3; an = 4  an1
 9.1.37: a1 = 2; an = n + an1
 9.1.38: a1 = 1; an = n  an1
 9.1.39: a1 = 5; an = 2an1
 9.1.40: a1 = 2; an = an1
 9.1.41: a1 = 3; an = an1 n
 9.1.42: a1 = 2; an = n + 3an1
 9.1.43: a1 = 1; a2 = 2; an = an1 # an2
 9.1.44: a1 = 1; a2 = 1; an = an2 + nan1
 9.1.45: a1 = A; an = an1 + d
 9.1.46: a1 = A; an = ran1 , r Z 0
 9.1.47: a1 = 22; an = 22 + an1
 9.1.48: a1 = 22; an = A an1 2
 9.1.49: n k=1 1k + 22
 9.1.50: n k=1 12k + 12
 9.1.51: n k=1 k2 2
 9.1.52: n k=1 1k + 122
 9.1.53: n k=0 1 3k
 9.1.54: n k=0 a 3 2 b k
 9.1.55: n1 k=0 1 3k+1
 9.1.56: n1 k=0 12k + 12
 9.1.57: n k=2 112k ln k
 9.1.58: n k=3 112k+1 2k
 9.1.59: 1 + 2 + 3 + + 20
 9.1.60: 13 + 23 + 33 + + 83
 9.1.61: 1 2 + 2 3 + 3 4 + + 13 13 + 1
 9.1.62: 1 + 3 + 5 + 7 + + 321122  14
 9.1.63: 1  1 3 + 1 9  1 27 + + 1126 1 36
 9.1.64: 2 3  4 9 + 8 27  + 11212 a 2 3 b 11
 9.1.65: 3 + 32 2 + 33 3 + + 3n n
 9.1.66: 1 e + 2 e2 + 3 e3 + + n en
 9.1.67: a + 1a + d2 + 1a + 2d2 + + 1a + nd2
 9.1.68: a + ar + ar2 + + arn1
 9.1.69: 40 k=1 5
 9.1.70: 50 k=1 8
 9.1.71: 40 k=1 k
 9.1.72: 24 k=1 1k2
 9.1.73: 20 k=1 15k + 32
 9.1.74: 26 k=1 13k  72
 9.1.75: 16 k=1 1k2 + 42
 9.1.76: 14 k=0 1k2  42
 9.1.77: 60 k=10 12k2
 9.1.78: a 40 k=8 13k2
 9.1.79: 20 k=5 k3
 9.1.80: 24 k=4 k3
 9.1.81: Credit Card Debt John has a balance of $3000 on his Discover card t...
 9.1.82: Trout Population A pond currently has 2000 trout in it. A fish hatc...
 9.1.83: Car Loans Phil bought a car by taking out a loan for $18,500 at 0.5...
 9.1.84: Environmental Control The Environmental Protection Agency (EPA) det...
 9.1.85: Growth of a Rabbit Colony A colony of rabbits begins with one pair ...
 9.1.86: Fibonacci Sequence Let define the nth term of a sequence. (a) Show ...
 9.1.87: Pascals Triangle Divide the triangular array shown (called Pascals ...
 9.1.88: Fibonacci Sequence Use the result of to do the following problems: ...
 9.1.89: Approximating f(x) ex In calculus, it can be shown that We can appr...
 9.1.90: Approximating f(x) ex Refer to 89. (a) Approximate with (b) Approxi...
 9.1.91: Bodes Law In 1772, Johann Bode published the following formula for ...
 9.1.92: Show that [Hint: Let Add these equations. Then S = n + 1n  12 + 1n...
 9.1.93: 25
 9.1.94: 28
 9.1.95: 221
 9.1.96: 289
 9.1.97: Triangular Numbers A triangular number is a term of the sequence Wr...
 9.1.98: For the sequence given in 97, show that un+1 = 1n + 121n + 222
 9.1.99: For the sequence given in 97, show that un+1 + un = 1n + 122
 9.1.100: Investigate various applications that lead to a Fibonacci sequence,...
 9.1.101: Write a paragraph that explains why
Solutions for Chapter 9.1: Sequences
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 9.1: Sequences
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321716811. Chapter 9.1: Sequences includes 101 full stepbystep solutions. Since 101 problems in chapter 9.1: Sequences have been answered, more than 36755 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.