 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: 4. Since 91 problems in chapter 5.1: Sequences have been answered, more than 45401 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 and is associated to the ISBN: 9780495391326.

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).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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 .

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.

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·

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

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