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INDEX
A
Absolute value of a real number, 71,
95, 101
Advice to the problem solver, v
Analogue of a formula, 16,
38, 39,
50
Angles of a polygon, 36
Arithmetic mean, 21, 24-27,
107-108, 111,
114, 116
Arithmetic progression, 20-21, 23-26,
34
Arrangements, 54-55, 59
Average (see Arithmetic mean)
Axiomatic approach, 28
B
Binomial, 3
Binomial coefficients, 2-10, 14-15,
24, 34,
37, 40-44,
46-54, 57,
115
Binomial expansion (see Binomial coefficients)
Binomial Theorem, 41, 43
C
Cauchy, Augustin (1789-1857), 110
Cauchy-Buniakowski Inequality, 110-117
Cauchy-Schwarz Inequality, 110-117
Closure, 101
Coefficients:
binomial, 2-10, 14-15,
24, 34,
37, 40-44,
46-54, 57,
115
complex, 60, 71
integral, 65-69
of linear equations, 76-77
multinomial, 53-57
of a polynomial, 60, 63,
65-69, 71-73,
96, 105
trinomial, 53-57
Cofactor, 81-83, 87,
91
Combinations, 51-59
Complex coefficients, 60, 71
Complex numbers, 1, 60-62,
70, 71
Consecutive integers, 33, 39
Consecutive terms of a progression, 21-24
Constructible lengths, 1
Convex polygons, 29, 36
Cramer's Rule, 77, 79,
82, 90
Crout's Method, 90
Cube roots, 22
D
Deduction, 28
Degree of a polynomial, 60, 66,
69, 70
Determinant, 74-100
Determinant of the system, 77-80,
90
Determinants of large order, 90
Diagonal:
of the Pascal Triangle, 4,
40
of a polygon, 29-31
of a square, 1
Difference:
of an arithmetic progression, 20-21,
23
of a function, 69-70
Discriminant of a quadratic, 105-106,
110
Division, synthetic 62-65, 70
Divisor, integral, 65, 66,
68
Double subscript notation, 81, 86
E
Elimination technique, 75, 79,
90
Equations:
linear, 74-83, 90,
99-100
polynomial, 38, 60-73,
106, 116
Even permutation, 56-58, 80-81,
86, 88-89
F
Factor Theorem, 61, 96
Factorials, 17-19, 34,
41-43, 46,
47, 51-55,
86
Factorizations, 26, 32-33,
36, 37,
61, 67-68,
71, 96-99
Fibonacci (c. 1170-1250), 11
Fibonacci numbers, 11-16, 27,
34, 37-39,
50, 94,
106, 117
Fibonacci Quarterly, The, 12
Fibonacci sequence (see Fibonacci numbers)
Final value in a summation, 45
Finite sequence, 20, 22
First-degree equation, 75
Fixed multiplier, 22
Fractions, 1
Function, 60, 69,
70-73, 105-106,
110
Fundamental Theorem of Algebra, 71
G
General polynomial, 45, 60
Geometric mean, 22, 25,
26, 107-108,
111, 114,
116
Geometric progression, 22-27, 34
H
Harmonic mean, 109, 114
Harmonic progression, 109
I
Identity, 44
Index of a summation, 45
Induction, 28-39, 41,
48, 69,
103, 104,
107, 115
Inequalities, 101-117
Initial value in a summation, 45
Integers, 1, 12,
13, 21,
26, 28-39,
41-49, 57,
58, 65-69,
102-104, 115-116
Integral coefficients, 65-69
Integral divisor, 65, 66,
68
Integral multiple, 26, 33,
36-37, 39,
48
Integral roots, 65-68
Irrational roots, 66-67, 69
L
Leonardo of Pisa (c. 1170-1250), 11
Liber Abacci, 11
Linear equations, 74-83, 90,
99-100
Lucas, Francois Edouard Anatole (1842-1891), 12
Lucas numbers, 12, 15-16,
34, 35,
38, 50,
94, 117
Lucas sequence(see Lucas numbers)
M
Mathematical induction, 28-39, 41,
48, 69,
103, 104,
107, 115
Mean:
arithmetic, 21, 24-27,
107-108, 111,
114, 116
geometric, 22, 25,
26, 107-108,
111, 114,
116
harmonic, 109, 114
Minor of a determinant, 81-83, 87,
91, 96
Multinomial coefficients, 53-57
Multiple, integral, 26, 33,
36-37, 39,
48
Multiple roots, 70
Multiplicity m, 70
Multiplier, fixed, 22
N
nth roots, 22, 101,
107
Number systems, 1
Numbers:
complex, 1, 60-62,
70, 71
Fibonacci, 11-16, 27,
34, 37-39,
50, 94,
106, 117
Lucas, 12, 15-16,
34, 35,
38, 50,
94, 117
natural, 1, 2
negative real, 1, 105-106,
107, 110
rational, 1, 66-69
real, 1, 22,
60, 101-117
triangular, 21, 31
Numerical approximation of a determinant, 90
O
Odd permutation, 56-58, 80-81,
86, 88-89
Order of a determinant, 75,
79, 87,
90
P
Pascal, Blaise (1623-1662), 3, 28
Pascal Triangle (see Binomial coefficients)
Pell sequence, 39
Permutations, 54-59, 80-81,
86, 88-89,
113
Pi notation, 46, 48,
115
Polygons, 29-31, 36
Polynomial equations, 38, 60-73,
106, 116
Polynomials, 18, 45,
48, 49,
60-73, 92,
96, 105-106,
110
Product notation, 46, 48,
115
Progression:
arithmetic, 20-21, 23-26,
34
geometric, 22-27, 34
harmonic, 109
Proper subset, 57
Q
Quadratic equation, 38, 106-107
Quadratic function, 105-107
Quotient, 22, 62,
65
R
R (before problem numbers), v
Ratio of a geometric progression, 22,
24, 25
Rational numbers, 1, 66-68
Rational roots, 66-68
Real numbers, 1, 22,
60, 101-117
Remainder in a division, 62-65
Remainder Theorem, 62
Roots:
cube, 22
integral, 65-68
irrational, 66-67, 69
multiple, 70
nth, 22, 101,
107
of a polynomial equation, 38,
60-73, 106,
116
rational, 66-68
square, 22, 104-105,
114, 115,
117
Row number of a binomial coefficient, 4
S
Schwarz, Hermann Amandus (1843-1921), 110
Schwarz Inequality, 110-117
Sequence:
arithmetic (progression), 20-21,
23-26, 34
factorial, 17-19
Fibonacci, 11-16, 27,
34, 37-39,
50, 94,
106, 117
geometric (progression), 22-27, 34
Lucas, 12, 15-16,
34, 35,
38, 50,
94, 117
miscellaneous, 26-27, 38,
49, 104
Pell, 39
Set, 26, 31,
51-53, 57
Sigma notation, 45-50, 69,
111
Simultaneous linear equations, 74-83,
90, 99-100
Square roots, 22, 104-105,
114, 115,
117
Strong mathematical induction, 34-35
Subsets, 51-53, 57
Sum:
of an arithmetic progression, 20,
23
of a geometric progression, 22,
24-26
of squares of binomial coefficients, 9
of squares of real numbers, 111
Summation notation, 45-50, 69,
111
Symmetric function, 71-73
Symmetry of the Pascal Triangle, 43,
53
Synthetic division, 62-65, 70
System of linear equations (see simultaneous linear equations)
Systems, number, 1
T
Theorems, deduction of, 28
Transpose of a determinant, 87
Transpose Theorem, 88, 91
Triangle, sides of a, 104, 116
Triangular numbers, 21, 31
Trichotomy, 101
Trinomial coefficients, 53-57
V
Vandermonde, Alexandre-Theophile (1735-1796), 96
Vandermonde determinants, 96-100
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Monday, August 3, 1998