 8.1.8.1.1: Fill in the blanks.An _______ is a function whose domain is the set...
 8.1.8.1.2: Fill in the blanks.The function values are called the _______ of a ...
 8.1.8.1.3: Fill in the blanks.A sequence is a ________ sequence if the domain ...
 8.1.8.1.4: Fill in the blanks.If you are given one or more of the first few te...
 8.1.8.1.5: Fill in the blanks.If n is a positive integer, n __ _____ is define...
 8.1.8.1.6: Fill in the blanks.The notation used to represent the sum of the te...
 8.1.8.1.7: Fill in the blanks.For the sum i is called the _______ of summation...
 8.1.8.1.8: Fill in the blanks.The sum of the terms of a finite or an infinite ...
 8.1.8.1.9: Fill in the blanks.The _______ of a sequence is the sum of the firs...
 8.1.8.1.10: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.11: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.12: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.13: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.14: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.15: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.16: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.17: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.18: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.19: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.20: In Exercises 120, write the first five terms of the sequence. (Assu...
 8.1.8.1.21: In Exercises 2126, find the indicated term of the sequence.
 8.1.8.1.22: In Exercises 2126, find the indicated term of the sequence.
 8.1.8.1.23: In Exercises 2126, find the indicated term of the sequence.
 8.1.8.1.24: In Exercises 2126, find the indicated term of the sequence.
 8.1.8.1.25: In Exercises 2126, find the indicated term of the sequence.
 8.1.8.1.26: In Exercises 2126, find the indicated term of the sequence.
 8.1.8.1.27: In Exercises 2732, use a graphing utility to graph the first 10 ter...
 8.1.8.1.28: In Exercises 2732, use a graphing utility to graph the first 10 ter...
 8.1.8.1.29: In Exercises 2732, use a graphing utility to graph the first 10 ter...
 8.1.8.1.30: In Exercises 2732, use a graphing utility to graph the first 10 ter...
 8.1.8.1.31: In Exercises 2732, use a graphing utility to graph the first 10 ter...
 8.1.8.1.32: In Exercises 2732, use a graphing utility to graph the first 10 ter...
 8.1.8.1.33: In Exercises 3338, use the table feature of a graphing utility to f...
 8.1.8.1.34: In Exercises 3338, use the table feature of a graphing utility to f...
 8.1.8.1.35: In Exercises 3338, use the table feature of a graphing utility to f...
 8.1.8.1.36: In Exercises 3338, use the table feature of a graphing utility to f...
 8.1.8.1.37: In Exercises 3338, use the table feature of a graphing utility to f...
 8.1.8.1.38: In Exercises 3338, use the table feature of a graphing utility to f...
 8.1.8.1.39: In Exercises 3942, match the sequence with its graph. [The graphs a...
 8.1.8.1.40: In Exercises 3942, match the sequence with its graph. [The graphs a...
 8.1.8.1.41: In Exercises 3942, match the sequence with its graph. [The graphs a...
 8.1.8.1.42: In Exercises 3942, match the sequence with its graph. [The graphs a...
 8.1.8.1.43: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.44: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.45: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.46: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.47: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.48: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.49: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.50: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.51: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.52: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.53: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.54: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.55: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.56: In Exercises 4356, write an expression for the apparent nth term of...
 8.1.8.1.57: In Exercises 5760, write the first five terms of the sequence defin...
 8.1.8.1.58: In Exercises 5760, write the first five terms of the sequence defin...
 8.1.8.1.59: In Exercises 5760, write the first five terms of the sequence defin...
 8.1.8.1.60: In Exercises 5760, write the first five terms of the sequence defin...
 8.1.8.1.61: In Exercises 61 64, write the first five terms of the sequence defi...
 8.1.8.1.62: In Exercises 61 64, write the first five terms of the sequence defi...
 8.1.8.1.63: In Exercises 61 64, write the first five terms of the sequence defi...
 8.1.8.1.64: In Exercises 61 64, write the first five terms of the sequence defi...
 8.1.8.1.65: In Exercises 6570, write the first five terms of the sequence. (Ass...
 8.1.8.1.66: In Exercises 6570, write the first five terms of the sequence. (Ass...
 8.1.8.1.67: In Exercises 6570, write the first five terms of the sequence. (Ass...
 8.1.8.1.68: In Exercises 6570, write the first five terms of the sequence. (Ass...
 8.1.8.1.69: In Exercises 6570, write the first five terms of the sequence. (Ass...
 8.1.8.1.70: In Exercises 6570, write the first five terms of the sequence. (Ass...
 8.1.8.1.71: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.72: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.73: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.74: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.75: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.76: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.77: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.78: In Exercises 7178, simplify the factorial expression.
 8.1.8.1.79: In Exercises 7990, find the sum.
 8.1.8.1.80: In Exercises 7990, find the sum.
 8.1.8.1.81: In Exercises 7990, find the sum.
 8.1.8.1.82: In Exercises 7990, find the sum.
 8.1.8.1.83: In Exercises 7990, find the sum.
 8.1.8.1.84: In Exercises 7990, find the sum.
 8.1.8.1.85: In Exercises 7990, find the sum.
 8.1.8.1.86: In Exercises 7990, find the sum.
 8.1.8.1.87: In Exercises 7990, find the sum.
 8.1.8.1.88: In Exercises 7990, find the sum.
 8.1.8.1.89: In Exercises 7990, find the sum.
 8.1.8.1.90: In Exercises 7990, find the sum.
 8.1.8.1.91: In Exercises 9194, use a graphing utility to find the sum.
 8.1.8.1.92: In Exercises 9194, use a graphing utility to find the sum.
 8.1.8.1.93: In Exercises 9194, use a graphing utility to find the sum.
 8.1.8.1.94: In Exercises 9194, use a graphing utility to find the sum.
 8.1.8.1.95: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.96: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.97: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.98: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.99: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.100: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.101: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.102: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.103: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.104: In Exercises 95104, use sigma notation to write the sum. Then use a...
 8.1.8.1.105: In Exercises 105108, find the indicated partial sum of the series.
 8.1.8.1.106: In Exercises 105108, find the indicated partial sum of the series.
 8.1.8.1.107: In Exercises 105108, find the indicated partial sum of the series.
 8.1.8.1.108: In Exercises 105108, find the indicated partial sum of the series.
 8.1.8.1.109: In Exercises 109112, find (a) the fourth partial sum and (b) the su...
 8.1.8.1.110: In Exercises 109112, find (a) the fourth partial sum and (b) the su...
 8.1.8.1.111: In Exercises 109112, find (a) the fourth partial sum and (b) the su...
 8.1.8.1.112: In Exercises 109112, find (a) the fourth partial sum and (b) the su...
 8.1.8.1.113: Compound Interest A deposit of $5000 is made in an account that ear...
 8.1.8.1.114: Compound Interest A deposit of $100 is made each month in an accoun...
 8.1.8.1.115: Fish A landlocked lake has been selected to be stocked in the year ...
 8.1.8.1.116: Tree Farm A tree farm in the year 2010 has 10,000 Douglas fir trees...
 8.1.8.1.117: Investment You decide to place $50 at the beginning of each month i...
 8.1.8.1.118: Mortgage Payments You borrow $150,000 at 9% interest compounded mon...
 8.1.8.1.119: Average Wages The average hourly wage rates (in dollars) for instru...
 8.1.8.1.120: Education The preprimary enrollments (in thousands) in the United S...
 8.1.8.1.121: Revenue The revenues (in millions of dollars) for California Pizza ...
 8.1.8.1.122: Sales The sales (in billions of dollars) for AnheuserBusch Companie...
 8.1.8.1.123: True or False? In Exercises 123 and 124, determine whether the stat...
 8.1.8.1.124: True or False? In Exercises 123 and 124, determine whether the stat...
 8.1.8.1.125: Fibonacci Sequence In Exercises 125 and 126, use the Fibonacci sequ...
 8.1.8.1.126: Fibonacci Sequence In Exercises 125 and 126, use the Fibonacci sequ...
 8.1.8.1.127: Use the table feature of a graphing utility to find the first five ...
 8.1.8.1.128: Do you recognize the terms of the sequence in Exercise 127? What se...
 8.1.8.1.129: Find an expression for and in terms of n.
 8.1.8.1.130: Use the result from Exercise 129 to show that Is this result the sa...
 8.1.8.1.131: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.132: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.133: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.134: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.135: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.136: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.137: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.138: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.139: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.140: In Exercises 131140, write the first five terms of the sequence.
 8.1.8.1.141: In Exercises 141146, write the first five terms of the sequence. Th...
 8.1.8.1.142: In Exercises 141146, write the first five terms of the sequence. Th...
 8.1.8.1.143: In Exercises 141146, write the first five terms of the sequence. Th...
 8.1.8.1.144: In Exercises 141146, write the first five terms of the sequence. Th...
 8.1.8.1.145: In Exercises 141146, write the first five terms of the sequence. Th...
 8.1.8.1.146: In Exercises 141146, write the first five terms of the sequence. Th...
 8.1.8.1.147: In Exercises 147150, find, if possible, (a) A B, (b) 2B 3A A , (c) ...
 8.1.8.1.148: In Exercises 147150, find, if possible, (a) A B, (b) 2B 3A A , (c) ...
 8.1.8.1.149: In Exercises 147150, find, if possible, (a) A B, (b) 2B 3A A , (c) ...
 8.1.8.1.150: In Exercises 147150, find, if possible, (a) A B, (b) 2B 3A A , (c) ...
Solutions for Chapter 8.1: Sequences, Series, and Probability
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 8.1: Sequences, Series, and Probability
Get Full SolutionsPrecalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Chapter 8.1: Sequences, Series, and Probability includes 150 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 150 problems in chapter 8.1: Sequences, Series, and Probability have been answered, more than 33080 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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 .

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).