 5.7.5.1.600: A list of numbers that are related to each other by a rule is calle...
 5.7.5.1.601: The numbers that form a sequence are called its .
 5.7.5.1.602: A sequence in which each term differs from the preceding term by a ...
 5.7.5.1.603: The amount by which each pair of successive terms differ in an arit...
 5.7.5.1.604: A sequence in which the ratio of any term to the term that directly...
 5.7.5.1.605: The constant found by dividing any term in a geometric sequence by ...
 5.7.5.1.606: In Exercises 714, write the first five terms of the arithmetic sequ...
 5.7.5.1.607: In Exercises 714, write the first five terms of the arithmetic sequ...
 5.7.5.1.608: In Exercises 714, write the first five terms of the arithmetic sequ...
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 5.7.5.1.610: In Exercises 714, write the first five terms of the arithmetic sequ...
 5.7.5.1.611: In Exercises 714, write the first five terms of the arithmetic sequ...
 5.7.5.1.612: In Exercises 714, write the first five terms of the arithmetic sequ...
 5.7.5.1.613: In Exercises 714, write the first five terms of the arithmetic sequ...
 5.7.5.1.614: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.615: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.616: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.617: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.618: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.619: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.620: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.621: In Exercises 1522, determine the indicated term for the arithmetic ...
 5.7.5.1.622: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.623: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.624: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.625: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.626: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.627: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.628: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.629: In Exercises 2330, write an expression for the general or nth term,...
 5.7.5.1.630: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.631: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.632: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.633: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.634: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.635: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.636: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.637: In Exercises 3138, determine the sum of the terms of the arithmetic...
 5.7.5.1.638: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.639: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.640: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.641: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.642: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.643: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.644: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.645: In Exercises 3946, write the first five terms of the geometric sequ...
 5.7.5.1.646: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.647: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.648: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.649: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.650: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.651: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.652: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.653: In Exercises 4754, determine the indicated term for the geometric s...
 5.7.5.1.654: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.655: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.656: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.657: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.658: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.659: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.660: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.661: In Exercises 5562, write an expression for the general or nth term,...
 5.7.5.1.662: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.663: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.664: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.665: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.666: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.667: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.668: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.669: In Exercises 6370, determine the sum of the first n terms of the ge...
 5.7.5.1.670: Determine the sum of the first 100 natural numbers.
 5.7.5.1.671: Determine the sum of the first 100 odd natural numbers.
 5.7.5.1.672: Determine the sum of the first 100 even natural numbers.
 5.7.5.1.673: Determine the sum of the first 50 multiples of 3.
 5.7.5.1.674: Pendulum Movement Each swing of a pendulum (from far left to far ri...
 5.7.5.1.675: Clock Strikes A clock strikes once at 1 oclock, twice at 2 oclock, ...
 5.7.5.1.676: Annual Pay Raises Rita Fernandez is given a starting salary of $35,...
 5.7.5.1.677: A Bouncing Ball Each time a ball bounces, the height attained by th...
 5.7.5.1.678: Enrollment Increase The current enrollment at Loras College is 8000...
 5.7.5.1.679: Samurai Sword Construction While making a traditional Japanese samu...
 5.7.5.1.680: Salary Increase If your salary were to increase at a rate of 6% per...
 5.7.5.1.681: A Bouncing Ball When dropped, a ball rebounds to fourfifths of its ...
 5.7.5.1.682: Value of a Stock Ten years ago, Nancy Hart purchased $2000 worth of...
 5.7.5.1.683: A Baseball Game During a baseball game, the visiting team scored 1 ...
 5.7.5.1.684: A geometric sequence has a1 = 82 and r = 1 2; find s6.
 5.7.5.1.685: Sums of Interior Angles The sums of the interior angles of a triang...
 5.7.5.1.686: Divisibility by 6 Determine how many numbers between 7 and 1610 are...
 5.7.5.1.687: Determine r and a1 for the geometric sequence with a2 = 24 and a5 =...
 5.7.5.1.688: Total Distance Traveled by a Bouncing Ball A ball is dropped from a...
 5.7.5.1.689: Martingale System See the Recreational Mathematics box on page 274....
 5.7.5.1.690: A topic generally associated with sequences is series. a) Research ...
Solutions for Chapter 5.7: Number Theory and the Real Number System
Full solutions for A Survey of Mathematics with Applications  9th Edition
ISBN: 9780321759665
Solutions for Chapter 5.7: Number Theory and the Real Number System
Get Full SolutionsSince 91 problems in chapter 5.7: Number Theory and the Real Number System have been answered, more than 79893 students have viewed full stepbystep solutions from this chapter. A Survey of Mathematics with Applications was written by and is associated to the ISBN: 9780321759665. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.7: Number Theory and the Real Number System includes 91 full stepbystep solutions. This textbook survival guide was created for the textbook: A Survey of Mathematics with Applications, edition: 9.

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.

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

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).

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

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

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.