 8.1: Determine whether the statement is true or false. A sequence is a f...
 8.2: Determine whether the statement is true or false. An infinite geome...
 8.3: Determine whether the statement is true or false. Permutations invo...
 8.4: Determine whether the statement is true or false. The total number ...
 8.5: Determine whether the statement is true or false. Find the first 4 ...
 8.6: Determine whether the statement is true or false. Predict the gener...
 8.7: Determine whether the statement is true or false. Find and evaluate...
 8.8: Determine whether the statement is true or false. Use a graphing ca...
 8.9: Determine whether the statement is true or false. Write sigma notat...
 8.10: Determine whether the statement is true or false. Find the 10th ter...
 8.11: Determine whether the statement is true or false. Find the 6th term...
 8.12: Determine whether the statement is true or false. Find the sum of t...
 8.13: Determine whether the statement is true or false. Find the sum of t...
 8.14: Determine whether the statement is true or false. The 1st term in a...
 8.15: Determine whether the statement is true or false. The common differ...
 8.16: Determine whether the statement is true or false. For a geometric s...
 8.17: Determine whether the statement is true or false. For a geometric s...
 8.18: Find the sum of each infinite geometric series, if itexists. 25 + 2...
 8.19: Find the sum of each infinite geometric series, if itexists. 0.27 +...
 8.20: Find the sum of each infinite geometric series, if itexists. 12 16...
 8.21: Find fraction notation for 2.43. [
 8.22: Insert four arithmetic means between 5 and 9.
 8.23: Bouncing Golfball. A golfball is dropped froma height of 30 ft to t...
 8.24: The Amount of an Annuity. To create acollege fund, a parent makes a...
 8.25: Total Gift. Suppose you receive 10 on the firstday of the year, 12 ...
 8.26: The Economic Multiplier. Suppose the governmentis making a $24,000,...
 8.27: Use mathematical induction to prove each of thefollowing. For every...
 8.28: Use mathematical induction to prove each of thefollowing. For every...
 8.29: Use mathematical induction to prove each of thefollowing. For every...
 8.30: Book Arrangements. In how many ways can6 books be arranged on a shelf?
 8.31: Flag Displays. If 9 different signal flags areavailable, how many d...
 8.32: Prize Choices. The winner of a contest canchoose any 8 of 15 prizes...
 8.33: FraternitySorority Names. The Greek alphabetcontains 24 letters. Ho...
 8.34: Letter Arrangements. In how many distinguishableways can the letter...
 8.35: Floor Plans. A manufacturer of houses has1 floor plan but achieves ...
 8.36: Code Symbols. How many code symbols can beformed using 5 out of 6 o...
 8.37: Determine the number of subsets of a set containing8 members.
 8.38: Expand. 1m + n27 1x  2225
 8.39: Expand. 1x  2225
 8.40: Expand. 1x2  3y24
 8.41: Expand. aa +1ab8
 8.42: Expand. 11 + 5i26, where i2 = 1
 8.43: Find the 4th term of 1a + x212.
 8.44: Find the 12th term of 12a  b218. Do not multiplyout the factorials.
 8.45: Rolling Dice. What is the probability of gettinga 10 on a roll of a...
 8.46: Drawing a Card. From a deck of 52 cards,1 card is drawn at random. ...
 8.47: Drawing Three Cards. From a deck of 52 cards,3 are drawn at random ...
 8.48: Election Poll. Three people were running formayor in an election ca...
 8.49: Consumption of American Cheese. The tablebelow lists the number of ...
 8.50: Which of the following is the 25th term of thearithmetic sequence 1...
 8.51: What is the probability of getting a total of 4 on aroll of a pair ...
 8.52: The graph of the sequence whose general term isan = n  1 is which ...
 8.53: Suppose that a1, a2, c, an and b1, b2, c, bn aregeometric sequences...
 8.54: Suppose that a1, a2, c, an is an arithmetic sequence.Is b1, b2, c, ...
 8.55: The zeros of this polynomial functionform an arithmetic sequence. F...
 8.56: Write the first 3 terms of the infinite geometricseries with r = 1...
 8.57: Simplify:a10k=0112ka10kb1log x210k1log y2k.
 8.58: Solve for n. an6b = 3 an  15b
 8.59: Solve for n. nn  1b = 36
 8.60: Solve for n. Solve for a:a5k=0a5kb95kak = 0.
 8.61: How long is 15? Suppose you own 15 booksand decide to make up all t...
 8.62: Circular Arrangements. In how many ways canthe numbers on a clock f...
 8.63: Give an explanation that you might use with afellow student to expl...
 8.64: Explain why a combination lock should reallybe called a permutation...
 8.65: Discuss the advantages and disadvantages of eachmethod of finding a...
Solutions for Chapter 8: Sequences, Series, and Combinatorics
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 8: Sequences, Series, and Combinatorics
Get Full SolutionsCollege Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. This textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions. Since 65 problems in chapter 8: Sequences, Series, and Combinatorics have been answered, more than 26149 students have viewed full stepbystep solutions from this chapter. Chapter 8: Sequences, Series, and Combinatorics includes 65 full stepbystep solutions.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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].

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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