 2.4.1E: Find these terms of (he sequence {an}. where an = 2 (?3)n + 5n.a) a...
 2.4.2E: What is the term as of the sequence {an} if an equals
 2.4.3E: What are the terms a0, a1, a2, and a3 of the sequence {an}, where a...
 2.4.4E: What are the terms a0, a1, a2, and a3 of the sequence {an} where an...
 2.4.5E: List the first 10 terms of each of these sequences.a) the sequence ...
 2.4.6E: List the first 10 terms of each of these sequences.a) the sequence ...
 2.4.7E: Find at least three different sequences beginning with the terms 1,...
 2.4.8E: Find at least three different sequences beginning with the terms 3,...
 2.4.9E: Find the first five terms of the sequence defined by each of these ...
 2.4.10E: Find the first six terms of the sequence defined by each of these r...
 2.4.11E: Let an = 2n + 5 3n for n = 0, 1, 2
 2.4.12E: Show that the sequence {an} is a solution of the recurrence relatio...
 2.4.13E: Is the sequence {an) a solution of the recurrence relation an = —8a...
 2.4.14E: For each of these sequences find a recurrence relation satisfied by...
 2.4.15E: Show that the sequence {an} is a solution of the recurrence relatio...
 2.4.16E: Find the solution to each of these recurrence relations with the gi...
 2.4.17E: Find the solution to each of these recurrence relations and initial...
 2.4.18E: A person deposits $1000 in an account that yields 9% interest compo...
 2.4.19E: Suppose that the number of bacteria in a colony triples every hour....
 2.4.20E: Assume that the population of the world in 2010 was 6.9 billion and...
 2.4.21E: A factory makes custom sports cars at an increasing rate. In the fi...
 2.4.22E: An employee joined a company in 2009 with a starting salary of $50....
 2.4.23E: Find a recurrence relation for the balance B(k) owed at the end of ...
 2.4.24E: a) Find a recurrence relation for the balance B(k) owed at the end ...
 2.4.25E: For each of these lists of integers, provide a simple formula or ru...
 2.4.26E: For each of these lists of integers, provide a simple formula or ru...
 2.4.27E: Show that if an denotes the nth positive integer that is not a perf...
 2.4.29E: What are the values of these sums?
 2.4.30E: What are the values of these sums, where S = {1, 3, 5, 7}?
 2.4.31E: What is the value of each of these sums of terms of a geometric pro...
 2.4.32E: Find the value of each of these sums.
 2.4.33E: Compute each of these double sums.
 2.4.34E: Compute each of these double sums.
 2.4.35E: Show that where is a sequence of real numbers. This type of sum is ...
 2.4.36E: Use the identity and Exercise 35 to compute .
 2.4.37E: Sum both sides of the identity k2 – (k – 1)2 = 2k – 1 from k = 1 to...
 2.4.38E: Use the technique given in Exercise 35. together with the result of...
 2.4.39E: Find Use Table 2.)
 2.4.40E: Find (Use Table 2.)
 2.4.41E: Find a formula for when m is a positive integer.
 2.4.42E: Find a formula for when m is a positive integer.There is also a spe...
 2.4.43E: What are the values of the following products? Recall that the valu...
 2.4.44E: Express n! using product notation.
 2.4.45E: Find
 2.4.46E: Find
Solutions for Chapter 2.4: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.4
Get Full SolutionsSince 45 problems in chapter 2.4 have been answered, more than 376874 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Chapter 2.4 includes 45 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

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!

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

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).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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

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

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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 .

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).

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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