 9.1.1: Discuss the meaning of a sequence. If a finite sequence is defined ...
 9.1.2: Describe three ways that a sequence can be defined.
 9.1.3: Is the ordered set of even numbers an infinite sequence? What about...
 9.1.4: What happens to the terms an of a sequence when there is a negative...
 9.1.5: What is a factorial, and how is it denoted? Use an example to illus...
 9.1.6: For the following exercises, write the first four terms of the sequ...
 9.1.7: For the following exercises, write the first four terms of the sequ...
 9.1.8: For the following exercises, write the first four terms of the sequ...
 9.1.9: For the following exercises, write the first four terms of the sequ...
 9.1.10: For the following exercises, write the first four terms of the sequ...
 9.1.11: For the following exercises, write the first four terms of the sequ...
 9.1.12: For the following exercises, write the first four terms of the sequ...
 9.1.13: For the following exercises, write the first four terms of the sequ...
 9.1.14: For the following exercises, write the first four terms of the sequ...
 9.1.15: For the following exercises, write the first four terms of the sequ...
 9.1.16: For the following exercises, write the first eight terms of the pie...
 9.1.17: For the following exercises, write the first eight terms of the pie...
 9.1.18: For the following exercises, write the first eight terms of the pie...
 9.1.19: For the following exercises, write the first eight terms of the pie...
 9.1.20: For the following exercises, write the first eight terms of the pie...
 9.1.21: For the following exercises, write an explicit formula for each seq...
 9.1.22: For the following exercises, write an explicit formula for each seq...
 9.1.23: For the following exercises, write an explicit formula for each seq...
 9.1.24: For the following exercises, write an explicit formula for each seq...
 9.1.25: For the following exercises, write an explicit formula for each seq...
 9.1.26: For the following exercises, write the first five terms of the sequ...
 9.1.27: For the following exercises, write the first five terms of the sequ...
 9.1.28: For the following exercises, write the first five terms of the sequ...
 9.1.29: For the following exercises, write the first five terms of the sequ...
 9.1.30: For the following exercises, write the first five terms of the sequ...
 9.1.31: For the following exercises, write the first eight terms of the seq...
 9.1.32: For the following exercises, write the first eight terms of the seq...
 9.1.33: For the following exercises, write the first eight terms of the seq...
 9.1.34: For the following exercises, write a recursive formula for each seq...
 9.1.35: For the following exercises, write a recursive formula for each seq...
 9.1.36: For the following exercises, write a recursive formula for each seq...
 9.1.37: For the following exercises, write a recursive formula for each seq...
 9.1.38: For the following exercises, write a recursive formula for each seq...
 9.1.39: For the following exercises, evaluate the factorial. 6!
 9.1.40: For the following exercises, evaluate the factorial. 12 ___ 6 !
 9.1.41: For the following exercises, evaluate the factorial. ___ 6!
 9.1.42: For the following exercises, evaluate the factorial. 100! ____ 99!
 9.1.43: For the following exercises, write the first four terms of the sequ...
 9.1.44: For the following exercises, write the first four terms of the sequ...
 9.1.45: For the following exercises, write the first four terms of the sequ...
 9.1.46: For the following exercises, write the first four terms of the sequ...
 9.1.47: For the following exercises, graph the first five terms of the indi...
 9.1.48: For the following exercises, graph the first five terms of the indi...
 9.1.49: For the following exercises, graph the first five terms of the indi...
 9.1.50: For the following exercises, graph the first five terms of the indi...
 9.1.51: For the following exercises, graph the first five terms of the indi...
 9.1.52: For the following exercises, write an explicit formula for the sequ...
 9.1.53: For the following exercises, write an explicit formula for the sequ...
 9.1.54: For the following exercises, write an explicit formula for the sequ...
 9.1.55: For the following exercises, write a recursive formula for the sequ...
 9.1.56: For the following exercises, write a recursive formula for the sequ...
 9.1.57: Follow these steps to evaluate a sequence defined recursively using...
 9.1.58: Follow these steps to evaluate a sequence defined recursively using...
 9.1.59: Follow these steps to evaluate a sequence defined recursively using...
 9.1.60: Follow these steps to evaluate a sequence defined recursively using...
 9.1.61: Follow these steps to evaluate a sequence defined recursively using...
 9.1.62: Follow these steps to evaluate a finite sequence defined by an expl...
 9.1.63: Follow these steps to evaluate a finite sequence defined by an expl...
 9.1.64: Follow these steps to evaluate a finite sequence defined by an expl...
 9.1.65: Follow these steps to evaluate a finite sequence defined by an expl...
 9.1.66: Follow these steps to evaluate a finite sequence defined by an expl...
 9.1.67: Consider the sequence defined by an = 6 8n. Is an = 421 a term in t...
 9.1.68: What term in the sequence an = n2 + 4n + 4 __________ 2(n + 2) has ...
 9.1.69: Find a recursive formula for the sequence 1, 0, 1, 1, 0, 1, 1, 0, 1...
 9.1.70: Calculate the first eight terms of the sequences an = (n + 2)! ____...
 9.1.71: Prove the conjecture made in the preceding exercise.
Solutions for Chapter 9.1: SEQUENCES AND THEIR NOTATIONS
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 9.1: SEQUENCES AND THEIR NOTATIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 1. Since 71 problems in chapter 9.1: SEQUENCES AND THEIR NOTATIONS have been answered, more than 32126 students have viewed full stepbystep solutions from this chapter. College Algebra was written by and is associated to the ISBN: 9781938168383. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.1: SEQUENCES AND THEIR NOTATIONS includes 71 full stepbystep solutions.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.