 8.4.1: List the first five statements in the sequence that can beobtained ...
 8.4.2: List the first five statements in the sequence that can beobtained ...
 8.4.3: List the first five statements in the sequence that can beobtained ...
 8.4.4: List the first five statements in the sequence that can beobtained ...
 8.4.5: Use mathematical induction to prove each of thefollowing. 2 + 4 + 6...
 8.4.6: Use mathematical induction to prove each of thefollowing. 4 + 8 + 1...
 8.4.7: Use mathematical induction to prove each of thefollowing. 1 + 5 + 9...
 8.4.8: Use mathematical induction to prove each of thefollowing. 3 + 6 + 9...
 8.4.9: Use mathematical induction to prove each of thefollowing. 2 + 4 + 8...
 8.4.10: Use mathematical induction to prove each of thefollowing. 2 2n
 8.4.11: Use mathematical induction to prove each of thefollowing. n 6 n + 1
 8.4.12: Use mathematical induction to prove each of thefollowing. 3n 6 3n+1
 8.4.13: Use mathematical induction to prove each of thefollowing. 2n 2n
 8.4.14: Use mathematical induction to prove each of thefollowing. 1 2+12 3+...
 8.4.15: Use mathematical induction to prove each of thefollowing. 11 2 3+12...
 8.4.16: Use mathematical induction to prove each of thefollowing. If x is a...
 8.4.17: The following formulas can be used to find sums ofpowers of natural...
 8.4.18: The following formulas can be used to find sums ofpowers of natural...
 8.4.19: The following formulas can be used to find sums ofpowers of natural...
 8.4.20: The following formulas can be used to find sums ofpowers of natural...
 8.4.21: Use mathematical induction to prove each of thefollowing. ni=1i1i +...
 8.4.22: Use mathematical induction to prove each of thefollowing. a1 +11b a...
 8.4.23: Use mathematical induction to prove each of thefollowing. The sum o...
 8.4.24: Solve. 2x  3y = 1,3x  4y = 3
 8.4.25: Investment. Martin received $104 in simpleinterest one year from th...
 8.4.26: Use mathematical induction to prove each of the following. The sum ...
 8.4.27: Use mathematical induction to prove each of the following. + y is a...
 8.4.28: Prove each of the following using mathematical induction.Do the bas...
 8.4.29: Prove each of the following using mathematical induction.Do the bas...
 8.4.30: Prove each of the following for any complex numbersz1, z2, c, zn, w...
 8.4.31: Prove each of the following for any complex numbersz1, z2, c, zn, w...
 8.4.32: The Tower of Hanoi Problem. There are threepegs on a board. On one ...
Solutions for Chapter 8.4: Mathematical Induction
Full solutions for College Algebra: Graphs and Models  5th Edition
ISBN: 9780321783950
Solutions for Chapter 8.4: Mathematical Induction
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra: Graphs and Models, edition: 5. College Algebra: Graphs and Models was written by and is associated to the ISBN: 9780321783950. Since 32 problems in chapter 8.4: Mathematical Induction have been answered, more than 28923 students have viewed full stepbystep solutions from this chapter. Chapter 8.4: Mathematical Induction includes 32 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

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 .

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).