 65.1: Autumn said that the answer to Example 2 could have been found by e...
 65.2: Sierra said that 8, , 16, , 32 is a geometric sequence with three g...
 65.3: In 314, determine whether each given sequence is geometric. If it i...
 65.4: In 314, determine whether each given sequence is geometric. If it i...
 65.5: In 314, determine whether each given sequence is geometric. If it i...
 65.6: In 314, determine whether each given sequence is geometric. If it i...
 65.7: In 314, determine whether each given sequence is geometric. If it i...
 65.8: In 314, determine whether each given sequence is geometric. If it i...
 65.9: In 314, determine whether each given sequence is geometric. If it i...
 65.10: In 314, determine whether each given sequence is geometric. If it i...
 65.11: In 314, determine whether each given sequence is geometric. If it i...
 65.12: In 314, determine whether each given sequence is geometric. If it i...
 65.13: In 314, determine whether each given sequence is geometric. If it i...
 65.14: In 314, determine whether each given sequence is geometric. If it i...
 65.15: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.16: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.17: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.18: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.19: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.20: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.21: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.22: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.23: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.24: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.25: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.26: In 1526, write the first five terms of each geometric sequence. a1 ...
 65.27: What is the 10th term of the geometric sequence 0.25, 0.5, 1, . . .?
 65.28: What is the 9th term of the geometric sequence 125, 25, 5, . . .?
 65.29: In a geometric sequence, a1 5 1 and a5 5 16. Find a9.
 65.30: The first term of a geometric sequence is 1 and the 4th term is 27....
 65.31: In a geometric sequence, a1 5 2 and a3 5 16. Find a6.
 65.32: In a geometric sequence, a3 5 1 and a7 5 9. Find a1.
 65.33: Find two geometric means between 6 and 93.75.
 65.34: Find three geometric means between 3 and .
 65.35: Find three geometric means between 8 and 2,592.
 65.36: If $1,000 was invested at 6% annual interest at the beginning of 20...
 65.37: Al invested $3,000 in a certificate of deposit that pays 5% interes...
 65.38: In a small town, a census is taken at the beginning of each year. T...
 65.39: It is estimated that the deer population in a park was increasing b...
 65.40: A car that cost $20,000 depreciated by 20% each year. Find the valu...
 65.41: A manufacturing company purchases a machine for $50,000. Each year ...
Solutions for Chapter 65: Geometric Sequences
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 65: Geometric Sequences
Get Full SolutionsAmsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Chapter 65: Geometric Sequences includes 41 full stepbystep solutions. Since 41 problems in chapter 65: Geometric Sequences have been answered, more than 30856 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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!).

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 .

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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