 8.8.1: In Exercises 14, write the first five terms of the sequence. (Assum...
 8.8.2: In Exercises 14, write the first five terms of the sequence. (Assum...
 8.8.3: In Exercises 14, write the first five terms of the sequence. (Assum...
 8.8.4: In Exercises 14, write the first five terms of the sequence. (Assum...
 8.8.5: In Exercises 58, write an expression for the apparent th term of th...
 8.8.6: In Exercises 58, write an expression for the apparent th term of th...
 8.8.7: In Exercises 58, write an expression for the apparent th term of th...
 8.8.8: In Exercises 58, write an expression for the apparent th term of th...
 8.8.9: In Exercises 9 and 10, write the first five terms of the sequence d...
 8.8.10: In Exercises 9 and 10, write the first five terms of the sequence d...
 8.8.11: In Exercises 1114, simplify the factorial expression.
 8.8.12: In Exercises 1114, simplify the factorial expression.
 8.8.13: In Exercises 1114, simplify the factorial expression.
 8.8.14: In Exercises 1114, simplify the factorial expression.
 8.8.15: In Exercises 1522, find the sum.
 8.8.16: In Exercises 1522, find the sum.
 8.8.17: In Exercises 1522, find the sum.
 8.8.18: In Exercises 1522, find the sum.
 8.8.19: In Exercises 1522, find the sum.
 8.8.20: In Exercises 1522, find the sum.
 8.8.21: In Exercises 1522, find the sum.
 8.8.22: In Exercises 1522, find the sum.
 8.8.23: In Exercises 2326, use sigma notation to write the sum. Then use a ...
 8.8.24: In Exercises 2326, use sigma notation to write the sum. Then use a ...
 8.8.25: In Exercises 2326, use sigma notation to write the sum. Then use a ...
 8.8.26: In Exercises 2326, use sigma notation to write the sum. Then use a ...
 8.8.27: In Exercises 2730, find (a) the fourth partial sum and (b) the sum ...
 8.8.28: In Exercises 2730, find (a) the fourth partial sum and (b) the sum ...
 8.8.29: In Exercises 2730, find (a) the fourth partial sum and (b) the sum ...
 8.8.30: In Exercises 2730, find (a) the fourth partial sum and (b) the sum ...
 8.8.31: Compound Interest A deposit of $2500 is made in an account that ear...
 8.8.32: Education The numbers of fulltime faculty (in thousands) employed ...
 8.8.33: In Exercises 3336, determine whether or not the sequence is arithme...
 8.8.34: In Exercises 3336, determine whether or not the sequence is arithme...
 8.8.35: In Exercises 3336, determine whether or not the sequence is arithme...
 8.8.36: In Exercises 3336, determine whether or not the sequence is arithme...
 8.8.37: In Exercises 3740, write the first five terms of the arithmetic seq...
 8.8.38: In Exercises 3740, write the first five terms of the arithmetic seq...
 8.8.39: In Exercises 3740, write the first five terms of the arithmetic seq...
 8.8.40: In Exercises 3740, write the first five terms of the arithmetic seq...
 8.8.41: In Exercises 41 44, write the first five terms of the arithmetic se...
 8.8.42: In Exercises 41 44, write the first five terms of the arithmetic se...
 8.8.43: In Exercises 41 44, write the first five terms of the arithmetic se...
 8.8.44: In Exercises 41 44, write the first five terms of the arithmetic se...
 8.8.45: In Exercises 45 and 46, find a formula for for the arithmetic seque...
 8.8.46: In Exercises 45 and 46, find a formula for for the arithmetic seque...
 8.8.47: In Exercises 47 50, find the partial sum. Use a graphing utility to...
 8.8.48: In Exercises 47 50, find the partial sum. Use a graphing utility to...
 8.8.49: In Exercises 47 50, find the partial sum. Use a graphing utility to...
 8.8.50: In Exercises 47 50, find the partial sum. Use a graphing utility to...
 8.8.51: Find the sum of the first 100 positive multiples of 5.
 8.8.52: Find the sum of the integers from 20 to 80 (inclusive).
 8.8.53: Job Offer The starting salary for an accountant is $34,000 with a g...
 8.8.54: Baling Hay In his first trip baling hay around a field, a farmer ma...
 8.8.55: In Exercises 55 58, determine whether or not the sequence is geomet...
 8.8.56: In Exercises 55 58, determine whether or not the sequence is geomet...
 8.8.57: In Exercises 55 58, determine whether or not the sequence is geomet...
 8.8.58: In Exercises 55 58, determine whether or not the sequence is geomet...
 8.8.59: In Exercises 59 62, write the first five terms of the geometric seq...
 8.8.60: In Exercises 59 62, write the first five terms of the geometric seq...
 8.8.61: In Exercises 59 62, write the first five terms of the geometric seq...
 8.8.62: In Exercises 59 62, write the first five terms of the geometric seq...
 8.8.63: In Exercises 63 66, write the first five terms of the geometric seq...
 8.8.64: In Exercises 63 66, write the first five terms of the geometric seq...
 8.8.65: In Exercises 63 66, write the first five terms of the geometric seq...
 8.8.66: In Exercises 63 66, write the first five terms of the geometric seq...
 8.8.67: In Exercises 6770, find the th term of the geometric sequence and f...
 8.8.68: In Exercises 6770, find the th term of the geometric sequence and f...
 8.8.69: In Exercises 6770, find the th term of the geometric sequence and f...
 8.8.70: In Exercises 6770, find the th term of the geometric sequence and f...
 8.8.71: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.72: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.73: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.74: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.75: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.76: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.77: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.78: In Exercises 7178, find the sum. Use a graphing utility to verify y...
 8.8.79: In Exercises 7982, find the sum of the infinite geometric series.
 8.8.80: In Exercises 7982, find the sum of the infinite geometric series.
 8.8.81: In Exercises 7982, find the sum of the infinite geometric series.
 8.8.82: In Exercises 7982, find the sum of the infinite geometric series.
 8.8.83: Depreciation A company buys a fleet of six vans for $120,000. Durin...
 8.8.84: Annuity A deposit of $75 is made at the beginning of each month in ...
 8.8.85: In Exercises 85 88, use mathematical induction to prove the formula...
 8.8.86: In Exercises 85 88, use mathematical induction to prove the formula...
 8.8.87: In Exercises 85 88, use mathematical induction to prove the formula...
 8.8.88: In Exercises 85 88, use mathematical induction to prove the formula...
 8.8.89: In Exercises 8992, find the sum using the formulas for the sums of ...
 8.8.90: In Exercises 8992, find the sum using the formulas for the sums of ...
 8.8.91: In Exercises 8992, find the sum using the formulas for the sums of ...
 8.8.92: In Exercises 8992, find the sum using the formulas for the sums of ...
 8.8.93: In Exercises 9396, write the first five terms of the sequence begin...
 8.8.94: In Exercises 9396, write the first five terms of the sequence begin...
 8.8.95: In Exercises 9396, write the first five terms of the sequence begin...
 8.8.96: In Exercises 9396, write the first five terms of the sequence begin...
 8.8.97: In Exercises 97100, find the binomial coefficient. Use a graphing u...
 8.8.98: In Exercises 97100, find the binomial coefficient. Use a graphing u...
 8.8.99: In Exercises 97100, find the binomial coefficient. Use a graphing u...
 8.8.100: In Exercises 97100, find the binomial coefficient. Use a graphing u...
 8.8.101: In Exercises 101104, use Pascals Triangle to find the binomial coef...
 8.8.102: In Exercises 101104, use Pascals Triangle to find the binomial coef...
 8.8.103: In Exercises 101104, use Pascals Triangle to find the binomial coef...
 8.8.104: In Exercises 101104, use Pascals Triangle to find the binomial coef...
 8.8.105: In Exercises 105110, use the Binomial Theorem to expand and simplif...
 8.8.106: In Exercises 105110, use the Binomial Theorem to expand and simplif...
 8.8.107: In Exercises 105110, use the Binomial Theorem to expand and simplif...
 8.8.108: In Exercises 105110, use the Binomial Theorem to expand and simplif...
 8.8.109: In Exercises 105110, use the Binomial Theorem to expand and simplif...
 8.8.110: In Exercises 105110, use the Binomial Theorem to expand and simplif...
 8.8.111: Numbers in a Hat Slips of paper numbered 1 through 14 are placed in...
 8.8.112: Aircraft Boarding Eight people are boarding an aircraft. Two have t...
 8.8.113: Course Schedule A college student is preparing a course schedule of...
 8.8.114: Telemarketing A telemarketing firm is making calls to prospective c...
 8.8.115: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.116: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.117: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.118: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.119: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.120: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.121: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.122: In Exercises 115122, evaluate the expression. Use a graphing utilit...
 8.8.123: In Exercises 123 and 124, find the number of distinguishable permut...
 8.8.124: In Exercises 123 and 124, find the number of distinguishable permut...
 8.8.125: Sports There are 10 bicyclists entered in a race. In how many diffe...
 8.8.126: Sports From a pool of seven juniors and eleven seniors, four cocap...
 8.8.127: Exam Questions A student can answer any 15 questions from a total o...
 8.8.128: Lottery In the Lotto Texas game, a player chooses six distinct numb...
 8.8.129: In Exercises 129 and 130, solve for n.
 8.8.130: In Exercises 129 and 130, solve for n.
 8.8.131: Apparel A man has five pairs of socks (no two pairs are the same co...
 8.8.132: Bookshelf Order A child returns a fivevolume set of books to a boo...
 8.8.133: Data Analysis A sample of college students, faculty members, and ad...
 8.8.134: Tossing a Die A sixsided die is rolled six times. What is the prob...
 8.8.135: Poker Hand Five cards are drawn from an ordinary deck of 52 playing...
 8.8.136: Drawing a Card You randomly select a card from a 52card deck. What...
 8.8.137: True or False? In Exercises 137 and 138, determine whether the stat...
 8.8.138: True or False? In Exercises 137 and 138, determine whether the stat...
 8.8.139: Writing In your own words, explain what makes a sequence (a) arithm...
 8.8.140: Think About It How do the two sequences differ?
 8.8.141: Graphical Reasoning The graphs of two sequences are shown below. Id...
 8.8.142: Population Growth Consider an idealized population with the charact...
 8.8.143: Writing Explain what a recursion formula is.
 8.8.144: Writing Explain why the terms of a geometric sequence of positive t...
 8.8.145: Think About It How do the expansions of and differ?
 8.8.146: The probability of an event must be a real number in what interval?...
 8.8.1: In Exercises 1 4, write the first five terms of the sequence
 8.8.2: In Exercises 1 4, write the first five terms of the sequence
 8.8.3: In Exercises 1 4, write the first five terms of the sequence
 8.8.4: In Exercises 1 4, write the first five terms of the sequence
 8.8.5: Simplify 11! 4!4! 7!
 8.8.6: Simplify n!n 1!. 11!
 8.8.7: Simplify 2n!n 1!. n!
 8.8.8: Write an expression for the apparent nth term of the sequence 2, 5,...
 8.8.9: In Exercises 9 and 10, find a formula for the nth term of the seque...
 8.8.10: In Exercises 9 and 10, find a formula for the nth term of the seque...
 8.8.11: Use sigma notation to write 231 1232 1 . . . 2312 1.a1
 8.8.12: Use sigma notation to write 12 18 132 1128 . . ..
 8.8.13: In Exercises 1315, find the sum.
 8.8.14: In Exercises 1315, find the sum.
 8.8.15: In Exercises 1315, find the sum.
 8.8.16: Use mathematical induction to prove the formula 6 9 . . . 3n 3nn 12 .
 8.8.17: Use the Binomial Theorem to expand and simplify 2a 5b4.
 8.8.18: In Exercises 1821, evaluate the expression.
 8.8.19: In Exercises 1821, evaluate the expression.
 8.8.20: In Exercises 1821, evaluate the expression.
 8.8.21: In Exercises 1821, evaluate the expression.
 8.8.22: Solve for n in 4 nP3 n1P4.
 8.8.23: How many distinct license plates can be issued consisting of one le...
 8.8.24: Four students are randomly selected from a class of 25 to answer qu...
 8.8.25: A card is drawn from a standard deck of 52 playing cards. Find the ...
 8.8.26: In 2006, six of the eleven mens basketball teams in the Big Ten Con...
 8.8.27: Two integers from 1 to 60 are chosen by a random number generator. ...
 8.8.28: A weather forecast indicates that the probability of snow is 75%. W...
Solutions for Chapter 8: Sequences, Series, and Probability
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 8: Sequences, Series, and Probability
Get Full SolutionsThis textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 8: Sequences, Series, and Probability includes 174 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Since 174 problems in chapter 8: Sequences, Series, and Probability have been answered, more than 44579 students have viewed full stepbystep solutions from this chapter.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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).

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

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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 matrix A.
A square matrix that has no inverse: det(A) = o.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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