 5.1.1: Write the first four terms of the sequences defined by the formulas...
 5.1.2: Write the first four terms of the sequences defined by the formulas...
 5.1.3: Write the first four terms of the sequences defined by the formulas...
 5.1.4: Write the first four terms of the sequences defined by the formulas...
 5.1.5: Write the first four terms of the sequences defined by the formulas...
 5.1.6: Write the first four terms of the sequences defined by the formulas...
 5.1.7: Let ak = 2k + 1 and bk = (k 1)3 + k + 2 for all integers k 0. Show ...
 5.1.8: Compute the first fifteen terms of each of the sequences in 8 and 9...
 5.1.9: Compute the first fifteen terms of each of the sequences in 8 and 9...
 5.1.10: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.11: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.12: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.13: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.14: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.15: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.16: Find explicit formulas for sequences of the form a1, a2, a3,... wit...
 5.1.17: Consider the sequence defined by an = 2n + (1)n 1 4 for all integer...
 5.1.18: Let a0 = 2, a1 = 3, a2 = 2, a3 = 1, a4 = 0, a5 = 1, and a6 = 2. Com...
 5.1.19: Compute the summations and products in 1928.
 5.1.20: Compute the summations and products in 1928.
 5.1.21: Compute the summations and products in 1928.
 5.1.22: Compute the summations and products in 1928.
 5.1.23: Compute the summations and products in 1928.
 5.1.24: Compute the summations and products in 1928.
 5.1.25: Compute the summations and products in 1928.
 5.1.26: Compute the summations and products in 1928.
 5.1.27: Compute the summations and products in 1928.
 5.1.28: Compute the summations and products in 1928.
 5.1.29: Write the summations in 2932 in expanded form.
 5.1.30: Write the summations in 2932 in expanded form.
 5.1.31: Write the summations in 2932 in expanded form.
 5.1.32: Write the summations in 2932 in expanded form.
 5.1.33: Evaluate the summations and products in 3336 for the indicated valu...
 5.1.34: Evaluate the summations and products in 3336 for the indicated valu...
 5.1.35: Evaluate the summations and products in 3336 for the indicated valu...
 5.1.36: Evaluate the summations and products in 3336 for the indicated valu...
 5.1.37: Rewrite 3739 by separating off the final term.
 5.1.38: Rewrite 3739 by separating off the final term.
 5.1.39: Rewrite 3739 by separating off the final term.
 5.1.40: Write each of 4042 as a single summation.
 5.1.41: Write each of 4042 as a single summation.
 5.1.42: Write each of 4042 as a single summation.
 5.1.43: Write each of 4352 using summation or product notation.
 5.1.44: Write each of 4352 using summation or product notation.
 5.1.45: Write each of 4352 using summation or product notation.
 5.1.46: Write each of 4352 using summation or product notation.
 5.1.47: Write each of 4352 using summation or product notation.
 5.1.48: Write each of 4352 using summation or product notation.
 5.1.49: Write each of 4352 using summation or product notation.
 5.1.50: Write each of 4352 using summation or product notation.
 5.1.51: Write each of 4352 using summation or product notation.
 5.1.52: Write each of 4352 using summation or product notation.
 5.1.53: Transform each of 53 and 54 by making the change of variable i = k ...
 5.1.54: Transform each of 53 and 54 by making the change of variable i = k ...
 5.1.55: Transform each of 5558 by making the change of variable j = i 1.
 5.1.56: Transform each of 5558 by making the change of variable j = i 1.
 5.1.57: Transform each of 5558 by making the change of variable j = i 1.
 5.1.58: Transform each of 5558 by making the change of variable j = i 1.
 5.1.59: Write each of 5961 as a single summation or product.
 5.1.60: Write each of 5961 as a single summation or product.
 5.1.61: Write each of 5961 as a single summation or product.
 5.1.62: Compute each of 6276. Assume the values of the variables are restri...
 5.1.63: Compute each of 6276. Assume the values of the variables are restri...
 5.1.64: Compute each of 6276. Assume the values of the variables are restri...
 5.1.65: Compute each of 6276. Assume the values of the variables are restri...
 5.1.66: Compute each of 6276. Assume the values of the variables are restri...
 5.1.67: Compute each of 6276. Assume the values of the variables are restri...
 5.1.68: Compute each of 6276. Assume the values of the variables are restri...
 5.1.69: Compute each of 6276. Assume the values of the variables are restri...
 5.1.70: Compute each of 6276. Assume the values of the variables are restri...
 5.1.71: Compute each of 6276. Assume the values of the variables are restri...
 5.1.72: Compute each of 6276. Assume the values of the variables are restri...
 5.1.73: Compute each of 6276. Assume the values of the variables are restri...
 5.1.74: Compute each of 6276. Assume the values of the variables are restri...
 5.1.75: Compute each of 6276. Assume the values of the variables are restri...
 5.1.76: Compute each of 6276. Assume the values of the variables are restri...
 5.1.77: a. Prove that n! + 2 is divisible by 2, for all integers n 2. b. Pr...
 5.1.78: Prove that for all nonnegative integers n and r with r + 1 n, n r +...
 5.1.79: Prove that if p is a prime number and r is an integer with 0 < r < ...
 5.1.80: Suppose a[1], a[2], a[3],..., a[m] is a onedimensional array and c...
 5.1.81: Use repeated division by 2 to convert (by hand) the integers in 818...
 5.1.82: Use repeated division by 2 to convert (by hand) the integers in 818...
 5.1.83: Use repeated division by 2 to convert (by hand) the integers in 818...
 5.1.84: Make a trace table to trace the action of Algorithm 5.1.1 on the in...
 5.1.85: Make a trace table to trace the action of Algorithm 5.1.1 on the in...
 5.1.86: Make a trace table to trace the action of Algorithm 5.1.1 on the in...
 5.1.87: Write an informal description of an algorithm (using repeated divis...
 5.1.88: Use the algorithm you developed for exercise 87 to convert the inte...
 5.1.89: Use the algorithm you developed for exercise 87 to convert the inte...
 5.1.90: Use the algorithm you developed for exercise 87 to convert the inte...
 5.1.91: Write a formal version of the algorithm you developed for exercise 87.
Solutions for Chapter 5.1: Sequences
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.1: Sequences
Get Full SolutionsChapter 5.1: Sequences includes 91 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Since 91 problems in chapter 5.1: Sequences have been answered, more than 24386 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

Column space C (A) =
space of all combinations of the columns of A.

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

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.