 5.7.1E: The formula is true for all integers n ? 1. Use this fact to solve ...
 5.7.2E: The formula is true for all real numbers r except r = 1 and for all...
 5.7.3E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.4E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.5E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.6E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.7E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.8E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.9E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.10E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.11E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.12E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.13E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.14E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.15E: A sequence is defined recursively. Use iteration to guess an explic...
 5.7.16E: Solve the recurrence relation obtained as the answer to exercise (c...
 5.7.17E: Solve the recurrence relation obtained as the answer to exercise(c)...
 5.7.18E: Suppose d is a fixed constant and ao, a1, a2, ... is a sequence tha...
 5.7.19E: A worker is promised a bonus if he can increase his productivity by...
 5.7.20E: A runner targets herself to improve her time on a certain course by...
 5.7.21E: Suppose r is a fixed constant and ao, a1, a2 ... is a sequence that...
 5.7.22E: As shown in Example, if a bank pays interest at a rate of i compoun...
 5.7.23E: Suppose the population of a country increases at a steady rate of 3...
 5.7.24E: A chain letter works as follows: One person sends a copy of the let...
 5.7.25E: A certain computer algorithm executes twice as many operations when...
 5.7.26E: A person saving for retirement makes an initial deposit of $1,000 t...
 5.7.27E: A person borrows $3,000 on a bank credit card at a nominal rate of ...
 5.7.28E: Use mathematical induction to verify the correctness of the formula...
 5.7.29E: Use mathematical induction to verify the correctness of the formula...
 5.7.30E: Use mathematical induction to verify the correctness of the formula...
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 5.7.36E: Use mathematical induction to verify the correctness of the formula...
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 5.7.39E: Use mathematical induction to verify the correctness of the formula...
 5.7.40E: Use mathematical induction to verify the correctness of the formula...
 5.7.41E: Use mathematical induction to verify the correctness of the formula...
 5.7.42E: Use mathematical induction to verify the correctness of the formula...
 5.7.43E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.44E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.45E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.46E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.47E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.48E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.49E: A sequence is defined recursively. (a) Use iteration to guess an ex...
 5.7.50E: Determine whether the given recursively defined sequence satisfies ...
 5.7.51E: Determine whether the given recursively defined sequence satisfies ...
 5.7.52E: A single line divides a plane into two regions. Two lines (by cross...
 5.7.53E: Compute for small values of n (up to about 5 or 6). Conjecture expl...
 5.7.54E: In economics the behavior of an economy from one period to another ...
Solutions for Chapter 5.7: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.7
Get Full SolutionsChapter 5.7 includes 54 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since 54 problems in chapter 5.7 have been answered, more than 56010 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).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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 .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 •

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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

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