 5.6.1: Find the first four terms of each of the recursively defined sequen...
 5.6.2: Find the first four terms of each of the recursively defined sequen...
 5.6.3: Find the first four terms of each of the recursively defined sequen...
 5.6.4: Find the first four terms of each of the recursively defined sequen...
 5.6.5: Find the first four terms of each of the recursively defined sequen...
 5.6.6: Find the first four terms of each of the recursively defined sequen...
 5.6.7: Find the first four terms of each of the recursively defined sequen...
 5.6.8: Find the first four terms of each of the recursively defined sequen...
 5.6.9: Let a0, a1, a2,... be defined by the formula an = 3n + 1, for all i...
 5.6.10: Let b0, b1, b2,... be defined by the formula bn = 4n , for all inte...
 5.6.11: Let c0, c1, c2,... be defined by the formula cn = 2n 1 for all inte...
 5.6.12: Let s0,s1,s2,... be defined by the formula sn = (1)n n! for all int...
 5.6.13: Let t0, t1, t2,... be defined by the formula tn = 2 + n for all int...
 5.6.14: Let d0, d1, d2,... be defined by the formula dn = 3n 2n for all int...
 5.6.15: For the sequence of Catalan numbers defined in Example 5.6.4, prove...
 5.6.16: Use the recurrence relation and values for the Tower of Hanoi seque...
 5.6.17: Tower of Hanoi with Adjacency Requirement: Suppose that in addition...
 5.6.18: Tower of Hanoi with Adjacency Requirement: Suppose the same situati...
 5.6.19: FourPole Tower of Hanoi: Suppose that the Tower of Hanoi problem h...
 5.6.20: Tower of Hanoi Poles in a Circle: Suppose that instead of being lin...
 5.6.21: Double Tower of Hanoi: In this variation of the Tower of Hanoi ther...
 5.6.22: Fibonacci Variation: A single pair of rabbits (male and female) is ...
 5.6.23: Fibonacci Variation: A single pair of rabbits (male and female) is ...
 5.6.24: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.25: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.26: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.27: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.28: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.29: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.30: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.31: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.32: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.33: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.34: In 2434, F0, F1, F2,... is the Fibonacci sequence.
 5.6.35: (For students who have studied calculus) Prove that lim n Fn+1 Fn e...
 5.6.36: (For students who have studied calculus) Define x0, x1, x2,... as f...
 5.6.37: Compound Interest: Suppose a certain amount of money is deposited i...
 5.6.38: Compound Interest: Suppose a certain amount of money is deposited i...
 5.6.39: With each step you take when climbing a staircase, you can move up ...
 5.6.40: A set of blocks contains blocks of heights 1, 2, and 4 centimeters....
 5.6.41: Use the recursive definition of summation, together with mathematic...
 5.6.42: Use the recursive definition of product, together with mathematical...
 5.6.43: Use the recursive definition of product, together with mathematical...
 5.6.44: The triangle inequality for absolute value states that for all real...
Solutions for Chapter 5.6: Defining Sequences Recursively
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 5.6: Defining Sequences Recursively
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Chapter 5.6: Defining Sequences Recursively includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 44 problems in chapter 5.6: Defining Sequences Recursively have been answered, more than 57948 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

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.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

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.

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.

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

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).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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