 12.3.1: In a(n) sequence the ratio of successive terms is a constant
 12.3.2: If 0r0 6 1, the sum of the geometric series a k=1 ark1 is .
 12.3.3: If a series does not converge, it is called a
 12.3.4: A geometric sequence may be defined recursively.
 12.3.5: In a geometric sequence the common ratio is always a positive number.
 12.3.6: For a geometric sequence with first term a1 and common ratio r, whe...
 12.3.7: In 716, show that each sequence is geometric. Then find the common ...
 12.3.8: In 716, show that each sequence is geometric. Then find the common ...
 12.3.9: In 716, show that each sequence is geometric. Then find the common ...
 12.3.10: In 716, show that each sequence is geometric. Then find the common ...
 12.3.11: In 716, show that each sequence is geometric. Then find the common ...
 12.3.12: In 716, show that each sequence is geometric. Then find the common ...
 12.3.13: In 716, show that each sequence is geometric. Then find the common ...
 12.3.14: In 716, show that each sequence is geometric. Then find the common ...
 12.3.15: In 716, show that each sequence is geometric. Then find the common ...
 12.3.16: In 716, show that each sequence is geometric. Then find the common ...
 12.3.17: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.18: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.19: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.20: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.21: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.22: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.23: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.24: In 1724, find the fifth term and the nth term of the geometric sequ...
 12.3.25: In 2530, find the indicated term of each geometric sequence.
 12.3.26: In 2530, find the indicated term of each geometric sequence.
 12.3.27: In 2530, find the indicated term of each geometric sequence.
 12.3.28: In 2530, find the indicated term of each geometric sequence.
 12.3.29: In 2530, find the indicated term of each geometric sequence.
 12.3.30: In 2530, find the indicated term of each geometric sequence.
 12.3.31: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.32: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.33: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.34: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.35: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.36: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.37: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.38: In 3138, find the nth term an of each geometric sequence. When give...
 12.3.39: In 39 44, find each sum.
 12.3.40: In 39 44, find each sum.
 12.3.41: In 39 44, find each sum.
 12.3.42: In 39 44, find each sum.
 12.3.43: In 39 44, find each sum.
 12.3.44: In 39 44, find each sum.
 12.3.45: For 4550, use a graphing utility to find the sum of each geometric ...
 12.3.46: For 4550, use a graphing utility to find the sum of each geometric ...
 12.3.47: For 4550, use a graphing utility to find the sum of each geometric ...
 12.3.48: For 4550, use a graphing utility to find the sum of each geometric ...
 12.3.49: For 4550, use a graphing utility to find the sum of each geometric ...
 12.3.50: For 4550, use a graphing utility to find the sum of each geometric ...
 12.3.51: In 5166, determine whether each infinite geometric series converges...
 12.3.52: In 5166, determine whether each infinite geometric series converges...
 12.3.53: In 5166, determine whether each infinite geometric series converges...
 12.3.54: In 5166, determine whether each infinite geometric series converges...
 12.3.55: In 5166, determine whether each infinite geometric series converges...
 12.3.56: In 5166, determine whether each infinite geometric series converges...
 12.3.57: In 5166, determine whether each infinite geometric series converges...
 12.3.58: In 5166, determine whether each infinite geometric series converges...
 12.3.59: In 5166, determine whether each infinite geometric series converges...
 12.3.60: In 5166, determine whether each infinite geometric series converges...
 12.3.61: In 5166, determine whether each infinite geometric series converges...
 12.3.62: In 5166, determine whether each infinite geometric series converges...
 12.3.63: In 5166, determine whether each infinite geometric series converges...
 12.3.64: In 5166, determine whether each infinite geometric series converges...
 12.3.65: In 5166, determine whether each infinite geometric series converges...
 12.3.66: In 5166, determine whether each infinite geometric series converges...
 12.3.67: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.68: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.69: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.70: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.71: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.72: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.73: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.74: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.75: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.76: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.77: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.78: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.79: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.80: In 67 80, determine whether the given sequence is arithmetic, geome...
 12.3.81: Find x so that x, x + 2, and x + 3 are consecutive terms of a geome...
 12.3.82: Find x so that x  1, x, and x + 2 are consecutive terms of a geome...
 12.3.83: If you have been hired at an annual salary of $18,000 and expect to...
 12.3.84: A new piece of equipment cost a company $15,000. Each year, for tax...
 12.3.85: Initially, a pendulum swings through an arc of 2 feet. On each succ...
 12.3.86: A ball is dropped from a height of 30 feet. Each time it strikes th...
 12.3.87: In an old fable, a commoner who had saved the kings life was told h...
 12.3.88: Look at the figure. What fraction of the square is eventually shade...
 12.3.89: Suppose that, throughout the U.S. economy, individuals spend 90% of...
 12.3.90: Refer to 89. Suppose that the marginal propensity to consume throug...
 12.3.91: One method of pricing a stock is to discount the stream of future d...
 12.3.92: Refer to 91. Suppose that a stock pays an annual dividend of $2.50 ...
 12.3.93: A rich man promises to give you $1000 on September 1, 2011. Each da...
 12.3.94: You are interviewing for a job and receive two offers: A: $20,000 t...
 12.3.95: Which of the following choices, A or B, results in more money? A: T...
 12.3.96: You have just signed a 7year professional football league contract...
 12.3.97: Suppose you were offered a job in which you would work 8 hours per ...
 12.3.98: Can a sequence be both arithmetic and geometric? Give reasons for y...
 12.3.99: Make up a geometric sequence. Give it to a friend and ask for its 2...
 12.3.100: Make up two infinite geometric series, one that has a sum and one t...
 12.3.101: . Describe the similarities and differences between geometric seque...
Solutions for Chapter 12.3: Geometric Sequences; Geometric Series
Full solutions for Precalculus Enhanced with Graphing Utilities  6th Edition
ISBN: 9780132854351
Solutions for Chapter 12.3: Geometric Sequences; Geometric Series
Get Full SolutionsChapter 12.3: Geometric Sequences; Geometric Series includes 101 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 101 problems in chapter 12.3: Geometric Sequences; Geometric Series have been answered, more than 59727 students have viewed full stepbystep solutions from this chapter. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351. This textbook survival guide was created for the textbook: Precalculus Enhanced with Graphing Utilities, edition: 6.

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

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

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

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