 6.1: In 16: a. Write a recursive formula for an. b. Is the sequence arit...
 6.2: In 16: a. Write a recursive formula for an. b. Is the sequence arit...
 6.3: In 16: a. Write a recursive formula for an. b. Is the sequence arit...
 6.4: In 16: a. Write a recursive formula for an. b. Is the sequence arit...
 6.5: In 16: a. Write a recursive formula for an. b. Is the sequence arit...
 6.6: In 16: a. Write a recursive formula for an. b. Is the sequence arit...
 6.7: In 712, write each series in sigma notation. 1 1 3 1 6 1 10 1 15 1 ...
 6.8: In 712, write each series in sigma notation. 2 1 5 1 8 1 11 1 14 1 ...
 6.9: In 712, write each series in sigma notation. 4 1 6 1 8 1 10 1 12 1 14
 6.10: In 712, write each series in sigma notation. 12 1 32 1 52 1 72 1 92...
 6.11: In 712, write each series in sigma notation. 1 2 2 1 3 2 4 1 5 2 6 1 7
 6.12: In 712, write each series in sigma notation. 1 2 1 1 2 2 1 1 2 3 1 ...
 6.13: In an arithmetic sequence, a1 5 6 and d 5 5. Write the first five t...
 6.14: In an arithmetic sequence, a1 5 6 and d 5 5. Find a30.
 6.15: In an arithmetic sequence, a1 5 5 and a4 5 23. Find a12.
 6.16: In an arithmetic sequence, a3 5 0 and a10 5 70. Find a1.
 6.17: In a geometric sequence, a1 5 2 and a2 5 5. Write the first five te...
 6.18: In a geometric sequence, a1 5 2 and a2 5 5. Find a10.
 6.19: In a geometric sequence, a1 5 2 and a4 5 128. Find a6.
 6.20: Write a recursive formula for the sequence 12, 20, 30, 42, 56, 70, ...
 6.21: Write the first five terms of a sequence if a1 5 1 and an 5 4an1 1 3.
 6.22: Find six arithmetic means between 1 and 36.
 6.23: Find three geometric means between 1 and 625.
 6.24: In an arithmetic sequence, a1 5 1, d 5 8. If an 5 89, find n.
 6.25: In a geometric sequence, a1 5 1 and a5 5 . Find r.
 6.26: Write as the sum of terms and find the sum.
 6.27: Write as the sum of terms and find the sum.
 6.28: To the nearest hundredth, find the value of 3e3
 6.29: a. Write, in sigma notation, the series 3 1 1.5 1 0.75 1 0.375 1 0....
 6.30: A grocer makes a display of canned tomatoes that are on sale. There...
 6.31: A retail store pays cashiers $20,000 a year for the first year of e...
 6.32: Ben started a job that paid $40,000 a year. Each year after the fir...
 6.33: The number of handshakes that are exchanged if every person in a ro...
Solutions for Chapter 6: Sequences And Series
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 6: Sequences And Series
Get Full SolutionsChapter 6: Sequences And Series includes 33 full stepbystep solutions. Since 33 problems in chapter 6: Sequences And Series have been answered, more than 29559 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.