 4.1.1: Prove that 1 23 + 1 34 + + 1 (n+1)(n+2) = n 2n+4 for every positive...
 4.1.2: Prove that 1 13 + 1 35 + 1 57 + + 1 (2n1)(2n+1) = n 2n+1 for every ...
 4.1.3: Prove that 1 14 + 1 47 + 1 710 + + 1 (3n2)(3n+1) = n 3n+1 for every...
 4.1.4: Find a formula for 2+4+6+ +2n for every positive integer n and then...
 4.1.5: Prove that 1 + 5 + 9 + + (4n 3) = n(2n 1) for every positive intege...
 4.1.6: Prove that 2 + 5 + 8 + + (3n 1) = n(3n + 1)/2 for every positive in...
 4.1.7: Prove the result in Result 4.5: For every positive integer n, 12 + ...
 4.1.8: Prove the result in Result 4.6: For every positive integer n, 13 +2...
 4.1.9: Prove that 1 + 3 + 32 + + 3n = (3n+1 1)/2 for every positive intege...
 4.1.10: Let r 2 be an integer. Prove that 1+r +r2 + +rn = rn+1 1 r1 for eve...
 4.1.11: Prove that 1 2 + 2 3 + 3 4 + + n(n + 1) = n(n+1)(n+2) 3 for every p...
 4.1.12: (a) Let k N. Prove that if k2 + k + 5 is even, then (k + 1)2 + (k +...
 4.1.13: Consider the expression (n + 2)(n + 3)/2, where n N. (a) Is (n + 2)...
 4.1.14: Square tiles, one foot on each side, are to be placed on a square s...
Solutions for Chapter 4.1: The Principle of Mathematical Induction
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 4.1: The Principle of Mathematical Induction
Get Full SolutionsChapter 4.1: The Principle of Mathematical Induction includes 14 full stepbystep solutions. Since 14 problems in chapter 4.1: The Principle of Mathematical Induction have been answered, more than 12246 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

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.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.