 11.4.1: Find S n for each geometric series described
 11.4.2: Find S n for each geometric series described
 11.4.3: Find the sum of each geometric series
 11.4.4: Find the sum of each geometric series
 11.4.5: Find the sum of each geometric series
 11.4.6: Find the sum of each geometric series.
 11.4.7: Find the sum of each geometric series.
 11.4.8: Find the sum of each geometric series.
 11.4.9: Find the sum of each geometric series.
 11.4.10: Find the sum of each geometric series.
 11.4.11: Find the sum of each geometric series.
 11.4.12: Find S n for each geometric series described
 11.4.13: Find S n for each geometric series described
 11.4.14: Find S n for each geometric series described
 11.4.15: Find S n for each geometric series described
 11.4.16: Find the indicated term for each geometric series described.
 11.4.17: Find the indicated term for each geometric series described.
 11.4.18: Find the indicated term for each geometric series described.
 11.4.19: Find the indicated term for each geometric series described.
 11.4.20: Find S n for each geometric series described.
 11.4.21: Find S n for each geometric series described.
 11.4.22: Find S n for each geometric series described.
 11.4.23: Find S n for each geometric series described.
 11.4.24: Find S n for each geometric series described.
 11.4.25: Find S n for each geometric series described.
 11.4.26: Find S n for each geometric series described.
 11.4.27: Find S n for each geometric series described.
 11.4.28: In the book Roots, author Alex Haley traced his family history back...
 11.4.29: There is a legend of a king who wanted to reward a boy for a good d...
 11.4.30: Find the sum of each geometric series
 11.4.31: Find the sum of each geometric series
 11.4.32: Find the sum of each geometric series
 11.4.33: Find the sum of each geometric series
 11.4.34: Find the indicated term for each geometric series described
 11.4.35: Find the indicated term for each geometric series described
 11.4.36: Find the indicated term for each geometric series described
 11.4.37: Find the indicated term for each geometric series described
 11.4.38: Find S n for each geometric series described
 11.4.39: Find S n for each geometric series described
 11.4.40: Find S n for each geometric series described
 11.4.41: Find S n for each geometric series described
 11.4.42: Find S n for each geometric series described
 11.4.43: Find S n for each geometric series described
 11.4.44: Find S n for each geometric series described
 11.4.45: Find S n for each geometric series described
 11.4.46: Find the sum of each geometric series
 11.4.47: Find the sum of each geometric series
 11.4.48: Find the sum of each geometric series
 11.4.49: Find the sum of each geometric series
 11.4.50: Find the sum of each geometric series
 11.4.51: Find the sum of each geometric series
 11.4.52: Find the indicated term for each geometric series described
 11.4.53: Find the indicated term for each geometric series described
 11.4.54: A certain water filtration system can remove 80% of the contaminant...
 11.4.55: Use a graphing calculator to find the sum of each geometric series
 11.4.56: Use a graphing calculator to find the sum of each geometric series
 11.4.57: Use a graphing calculator to find the sum of each geometric series
 11.4.58: Use a graphing calculator to find the sum of each geometric series
 11.4.59: Write a geometric series for which r = _1 2 and n = 4.
 11.4.60: Explain how to write the series 2 + 12 + 72 + 432 + 2592 using sigm...
 11.4.61: Explain how to write the series 2 + 12 + 72 + 432 + 2592 using sigm...
 11.4.62: Explain, using geometric series, why the polynomial 1 + x + x 2 + x...
 11.4.63: Use the information on page 643 to explain how emailing a joke is ...
 11.4.64: The first term of a geometric series is 1, and the common ratio is...
 11.4.65: Which set of dimensions corresponds to a rectangle similar to the o...
 11.4.66: Find the geometric means in each sequence
 11.4.67: Find the geometric means in each sequence
 11.4.68: Find the sum of each arithmetic series
 11.4.69: Find the sum of each arithmetic series
 11.4.70: Solve each equation. Check your solutions.
 11.4.71: Solve each equation. Check your solutions.
 11.4.72: Determine whether each graph represents an odddegree polynomial fu...
 11.4.73: Determine whether each graph represents an odddegree polynomial fu...
 11.4.74: Factor completely. If the polynomial is not factorable, write prime.
 11.4.75: Factor completely. If the polynomial is not factorable, write prime.
 11.4.76: Factor completely. If the polynomial is not factorable, write prime.
 11.4.77: For Exercises 7779, use the table that shows the percent of the Cal...
 11.4.78: For Exercises 7779, use the table that shows the percent of the Cal...
 11.4.79: For Exercises 7779, use the table that shows the percent of the Cal...
 11.4.80: Evaluate _a 1  b for the given values of a and b.
 11.4.81: Evaluate _a 1  b for the given values of a and b.
 11.4.82: Evaluate _a 1  b for the given values of a and b.
 11.4.83: Evaluate _a 1  b for the given values of a and b.
 11.4.84: Evaluate _a 1  b for the given values of a and b.
 11.4.85: Evaluate _a 1  b for the given values of a and b.
Solutions for Chapter 11.4: Geometric Series
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 11.4: Geometric Series
Get Full SolutionsSince 85 problems in chapter 11.4: Geometric Series have been answered, more than 47789 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. Chapter 11.4: Geometric Series includes 85 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

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.)

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.

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].

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 matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.