 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 45354 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: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

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

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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