 8.2.6E: How many different messages can be transmitted in n microseconds us...
 8.2.3E: Solve these recurrence relations together with the initial conditio...
 8.2.2E: Determine which of these are linear homogeneous recurrence relation...
 8.2.5E: How many different messages can be transmitted in n microseconds us...
 8.2.4E: Solve these recurrence relations together with the initial conditio...
 8.2.1E: Determine which of these are linear homogeneous recurrence relation...
 8.2.7E: In how many ways can a 2 × n rectangular checkerboard be tiled usin...
 8.2.8E: A model for the number of lobsters caught per year is based on the ...
 8.2.9E: A deposit of $100,000 is made to an investment fund at the beginnin...
 8.2.10E: Prove Theorem 2.
 8.2.11E: The Lucas numbers satisfy the recurrence relation Ln = Ln?1 + Ln?2,...
 8.2.12E: Find the solution to an = 2an?1 +an?2 ? 2an?3 for n = 3,4, 5,..., w...
 8.2.14E: Find the solution to an = 5an?2 ? 4an?4 with a0 = 3, a1 = 2, a2 = 6...
 8.2.13E: Find the solution to an = 7an?2 + 6an?3 with a0 = 9, a1 = 10, and a...
 8.2.15E: Find the solution to an = 2an?1 + 5an?2 ? 6an?3 with a0 = 7, a1 = ?...
 8.2.16E: Prove Theorem 3.
 8.2.17E: Prove this identity relating the Fibonacci numbers and the binomial...
 8.2.18E: Solve the recurrence relation an = 6an?1 ? 12an?2 + 8an?3 with a0 =...
 8.2.19E: Solve the recurrence relation an = ?3an?1 ? 3an?2 ? an?3 with a0 = ...
 8.2.20E: Find the general form of the solutions of the recurrence relation a...
 8.2.21E: What is the general form of the solutions of a linear homogeneous r...
 8.2.22E: What is the general form of the solutions of a linear homogeneous r...
 8.2.23E: Consider the nonhomogeneous linear recurrence relation an = 3an?1 +...
 8.2.25E: a) Determine values of the constants A and B such that an = An + B ...
 8.2.26E: What is the general form of the particular solution guaranteed to e...
 8.2.24E: Consider the nonhomogeneous linear recurrence relation an = 2an?1 +...
 8.2.27E: What is the general form of the particular solution guaranteed to e...
 8.2.28E: a) Find all solutions of the recurrence relation an = 2an?1 + 2n2._...
 8.2.29E: a) Find all solutions of the recurrence relation an = 2an?1 + 3n.__...
 8.2.30E: a) Find all solutions of the recurrence relation an = ?5an?1 ? 6an?...
 8.2.31E: Find all solutions of the recurrence relation an = 5an?1 ? 6an?2 + ...
 8.2.32E: Find the solution of the recurrence relation an = 2an?1 +3?2n.
 8.2.33E: Find all solutions of the recurrence relation an = 4an?1 ? 4an?2 + ...
 8.2.34E: Find all solutions of the recurrence relation an = 7an?1 ? 16an?2 +...
 8.2.35E: Find the solution of the recurrence relation an = 4an?1 ? 3an?2 + 2...
 8.2.36E: Let an be the sum of the first n triangular numbers, that is, where...
 8.2.38E: a) Find the characteristic roots of the linear homogeneous recurren...
 8.2.39E: a) Find the characteristic roots of the linear homogeneous recurren...
 8.2.40E: Solve the simultaneous recurrence relations with a0 = 1 and b0 = 2.
 8.2.41E: a) Use the formula found in Example 4 for fnthe nth Fibonacci numbe...
 8.2.42E: Show that if an = an?1 + an?2 , a0 = s and a1 = t, where s and t ar...
 8.2.43E: Express the solution of the linear nonhomogenous recurrence relatio...
 8.2.45E: Suppose that each pair of a genetically engineered species of rabbi...
 8.2.44E: (Linear algebra required) Let An be the n × n matrix with 2s on its...
 8.2.46E: Suppose that there are two goats on an island initially. The number...
 8.2.47E: A new employee at an exciting new software company starts with a sa...
 8.2.48E: a) Show that the recurrence relationf(n)an = g(n)an?1 + h(n),for n ...
 8.2.49E: Use Exercise 48 to solve the recurrence relation(n + 1)an = (n + 3)...
 8.2.50E: It can be shown that Cn,, the average number of comparisons made by...
 8.2.51E: Prove Theorem 4.
 8.2.52E: Prove Theorem 6.
 8.2.53E: Solve the recurrence relation T(n)=nT2(n/2) with initial condition ...
Solutions for Chapter 8.2: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 8.2
Get Full SolutionsSince 52 problems in chapter 8.2 have been answered, more than 185857 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Chapter 8.2 includes 52 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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).

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

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

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