 5.2.1: Use mathematical induction (and the proof of Proposition 5.2.1 as a...
 5.2.2: Use mathematical induction to show that any postage of at least 12c...
 5.2.3: For each positive integer n, let P(n) be the formula 12 + 22 ++ n2 ...
 5.2.4: For each integer n with n 2, let P(n) be the formula n1 i=1 i(i + 1...
 5.2.5: Fill in the missing pieces in the following proof that 1 + 3 + 5 ++...
 5.2.6: Prove each statement in 69 using mathematical induction. Do not der...
 5.2.7: Prove each statement in 69 using mathematical induction. Do not der...
 5.2.8: Prove each statement in 69 using mathematical induction. Do not der...
 5.2.9: Prove each statement in 69 using mathematical induction. Do not der...
 5.2.10: Prove each of the statements in 1017 by mathematical induction.
 5.2.11: Prove each of the statements in 1017 by mathematical induction.
 5.2.12: Prove each of the statements in 1017 by mathematical induction.
 5.2.13: Prove each of the statements in 1017 by mathematical induction.
 5.2.14: Prove each of the statements in 1017 by mathematical induction.
 5.2.15: Prove each of the statements in 1017 by mathematical induction.
 5.2.16: Prove each of the statements in 1017 by mathematical induction.
 5.2.17: Prove each of the statements in 1017 by mathematical induction.
 5.2.18: If x is a real number not divisible by , then for all integers n 1,...
 5.2.19: (For students who have studied calculus) Use mathematical induction...
 5.2.20: Use the formula for the sum of the first n integers and/or the form...
 5.2.21: Use the formula for the sum of the first n integers and/or the form...
 5.2.22: Use the formula for the sum of the first n integers and/or the form...
 5.2.23: Use the formula for the sum of the first n integers and/or the form...
 5.2.24: Use the formula for the sum of the first n integers and/or the form...
 5.2.25: Use the formula for the sum of the first n integers and/or the form...
 5.2.26: Use the formula for the sum of the first n integers and/or the form...
 5.2.27: Use the formula for the sum of the first n integers and/or the form...
 5.2.28: Use the formula for the sum of the first n integers and/or the form...
 5.2.29: Use the formula for the sum of the first n integers and/or the form...
 5.2.30: Find a formula in n, a, m, and d for the sum (a + md) + (a + (m + 1...
 5.2.31: Find a formula in a,r, m, and n for the sum ar m + ar m+1 + ar m+2 ...
 5.2.32: You have two parents, four grandparents, eight greatgrandparents, a...
 5.2.33: Find the mistakes in the proof fragments in 3335.
 5.2.34: Find the mistakes in the proof fragments in 3335.
 5.2.35: Find the mistakes in the proof fragments in 3335.
 5.2.36: Use Theorem 5.2.2 to prove that if m and n are any positive integer...
 5.2.37: Use Theorem 5.2.2 and the result of exercise 10 to prove that if p ...
Solutions for Chapter 5.2: Mathematical Induction I
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.2: Mathematical Induction I
Get Full SolutionsSince 37 problems in chapter 5.2: Mathematical Induction I have been answered, more than 24417 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: Mathematical Induction I includes 37 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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

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

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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 matrix N.
The columns of N are the n  r special solutions to As = O.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.