- 9.4.1: 2 + 4 + 6 + + 2n = n1n + 12
- 9.4.2: 3 + 5 + 7 + + 12n + 12 = n1n + 22
- 9.4.3: 3 + 4 + 5 + + 1n + 22 = 1 2 n1n + 52
- 9.4.4: 3 + 5 + 7 + + 12n + 12 = n1n + 22
- 9.4.5: 2 + 5 + 8 + + 13n - 12 = 1 2 n13n + 12
- 9.4.6: 1 + 4 + 7 + + 13n - 22 = 1 2 n13n - 12
- 9.4.7: 1 + 2 + 22 + + 2n-1 = 2n - 1
- 9.4.8: 1 + 3 + 32 + + 3n-1 = 1 2 13n - 12
- 9.4.9: 1 + 4 + 42 + + 4n-1 = 1 3 14n - 12
- 9.4.10: 1 + 5 + 52 + + 5n-1 = 1 4 15n - 12
- 9.4.11: 1 1 # 2 + 1 2 # 3 + 1 3 # 4 + + 1 n1n + 12 = n n + 1
- 9.4.12: 1 1 # 3 + 1 3 # 5 + 1 5 # 7 + + 1 12n - 1212n + 12 = n 2n + 1
- 9.4.13: 12 + 22 + 32 + + n2 = 1 6 n1n + 1212n + 12
- 9.4.14: 13 + 23 + 33 + + n3 = 1 4 n2 1n + 122
- 9.4.15: 4 + 3 + 2 + + 15 - n2 = 1 2 n19 - n2
- 9.4.16: -2 - 3 - 4 - - 1n + 12 = - 1 2 n1n + 32
- 9.4.17: 1 # 2 + 2 # 3 + 3 # 4 + + n1n + 12 = 1 3 n1n + 121n + 22
- 9.4.18: 1 # 2 + 3 # 4 + 5 # 6 + + 12n - 1212n2 = 1 3 n1n + 1214n - 12
- 9.4.19: n is divisible by 2
- 9.4.20: n is divisible by 3.
- 9.4.21: n is divisible by 2.
- 9.4.22: n1n + 121n + 22 is divisible by 6.
- 9.4.23: If then xn x 7 1, 7 1.
- 9.4.24: If then xn 0 6 x 6 1, 6 1.
- 9.4.25: a - b is a factor of an - bn
- 9.4.26: a + b is a factor of a2n+1 + b2n+1
- 9.4.27: (1 + a) n 1 + na, for a 7 0
- 9.4.28: Show that the statement is a prime number is true for but is not tr...
- 9.4.29: Show that the formula 2 + 4 + 6 + + 2n = n2 + n + 2 obeys Condition...
- 9.4.30: Use mathematical induction to prove that if then a + ar + ar2 + + a...
- 9.4.31: Use mathematical induction to prove that a + 1a + d2 + 1a + 2d2 + +...
- 9.4.32: Extended Principle of Mathematical Induction The Extended Principle...
- 9.4.33: Geometry Use the Extended Principle of Mathematical Induction to sh...
- 9.4.34: How would you explain the Principle of Mathematical Induction to a ...
Solutions for Chapter 9.4: Mathematical Induction
Full solutions for College Algebra | 9th Edition
ISBN: 9780321716811
Solutions for Chapter 9.4: Mathematical Induction
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Key Math Terms and definitions covered in this textbook
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Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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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].
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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.
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Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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Iterative method.
A sequence of steps intended to approach the desired solution.
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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.
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Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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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.
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Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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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.
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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.
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Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.