 5.3.1: Based on the discussion of the product 1 1 2 1 1 3 1 1 4 1 1 n at t...
 5.3.2: Experiment with computing values of the product 1 + 1 1 1 + 1 2 1 +...
 5.3.3: Observe that 1 13 = 1 3 1 13 + 1 35 = 2 5 1 13 + 1 35 + 1 57 = 3 7 ...
 5.3.4: Observe that 1 = 1, 1 4 = (1 + 2), 1 4 + 9 = 1 + 2 + 3, 1 4 + 9 16 ...
 5.3.5: Evaluate the sum n k=1 k (k + 1)! for n = 1, 2, 3, 4, and 5. Make a...
 5.3.6: For each positive integer n, let P(n) be the property 5n 1 is divis...
 5.3.7: For each positive integer n, let P(n) be the property 2n < (n + 1)!...
 5.3.8: Prove each statement in 823 by mathematical induction.
 5.3.9: Prove each statement in 823 by mathematical induction.
 5.3.10: Prove each statement in 823 by mathematical induction.
 5.3.11: Prove each statement in 823 by mathematical induction.
 5.3.12: Prove each statement in 823 by mathematical induction.
 5.3.13: Prove each statement in 823 by mathematical induction.
 5.3.14: Prove each statement in 823 by mathematical induction.
 5.3.15: Prove each statement in 823 by mathematical induction.
 5.3.16: Prove each statement in 823 by mathematical induction.
 5.3.17: Prove each statement in 823 by mathematical induction.
 5.3.18: Prove each statement in 823 by mathematical induction.
 5.3.19: Prove each statement in 823 by mathematical induction.
 5.3.20: Prove each statement in 823 by mathematical induction.
 5.3.21: Prove each statement in 823 by mathematical induction.
 5.3.22: Prove each statement in 823 by mathematical induction.
 5.3.23: Prove each statement in 823 by mathematical induction.
 5.3.24: A sequence a1, a2, a3,... is defined by letting a1 = 3 and ak = 7ak...
 5.3.25: A sequence b0, b1, b2,... is defined by letting b0 = 5 and bk = 4 +...
 5.3.26: A sequence b0, b1, b2,... is defined by letting b0 = 5 and bk = 4 +...
 5.3.27: A sequence d1, d2, d3,... is defined by letting d1 = 2 and dk = dk1...
 5.3.28: Prove that for all integers n 1, 1 3 = 1 + 3 5 + 7 = 1 + 3 + 5 7 + ...
 5.3.29: Prove that for all integers n 1, 1 3 = 1 + 3 5 + 7 = 1 + 3 + 5 7 + ...
 5.3.30: Theorem: For any integer n 1, all the numbers in a set of n numbers...
 5.3.31: For all integers n 1, 3n 2 is even. Proof (by mathematical inductio...
 5.3.32: Some 5 5 checkerboards with one square removed can be completely co...
 5.3.33: Consider a 4 6 checkerboard. Draw a covering of the board by Lshap...
 5.3.34: a. Use mathematical induction to prove that any checkerboard with d...
 5.3.35: Let m and n be any integers that are greater than or equal to 1. a....
 5.3.36: In a roundrobin tournament each team plays every other team exactl...
 5.3.37: On the outside rim of a circular disk the integers from 1 through 3...
 5.3.38: Suppose that n as and n bs are distributed around the outside of a ...
 5.3.39: For a polygon to be convex means that all of its interior angles ar...
 5.3.40: a. Prove that in an 8 8 checkerboard with alternating black and whi...
Solutions for Chapter 5.3: Mathematical Induction II
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.3: Mathematical Induction II
Get Full SolutionsSince 40 problems in chapter 5.3: Mathematical Induction II have been answered, more than 43598 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Chapter 5.3: Mathematical Induction II includes 40 full stepbystep solutions. 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).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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.

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.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.