A Primer on Number Sequences by Shailesh Shirali Mathematical Marvels Universities Press

Title Page......Page 4
Copyright......Page 5
Preface......Page 12
Part A The Method of Differences......Page 15
1.1 What is a pattern?......Page 16
1.2 Number sequences......Page 21
1.3 Sequences in mathematics......Page 22
2 Terms and Symbols......Page 28
3.1 Introduction......Page 41
Exercises......Page 42
3.2 A symbolic apparatus......Page 45
Exercises......Page 46
4.1 Introduction......Page 48
4.2 A worked example......Page 49
4.3 Warning!......Page 52
4.4 Interlude: Hypotheses in science......Page 53
Exercises......Page 55
4.5 Triangular numbers......Page 56
4.6 Diagonals of a polygon......Page 59
4.7 Pentagonal numbers......Page 61
4.8 Sums of squares......Page 63
4.9 Proof of the formula......Page 65
Exercises......Page 66
5.1.2 The Sequence {2n}......Page 68
5.1.3 The Sequence {3n}......Page 69
5.2 Mixed sequences......Page 70
5.3 Seeking an explanation......Page 71
Exercises......Page 72
6.1 General comments......Page 77
Exercises......Page 80
List of references......Page 84
Part B A Gallery of Sequences Introduction......Page 86
7.1 Generating the squares......Page 88
7.2 Last digit......Page 89
7.3 Coin changing using squares......Page 90
7.4 Sums of squares......Page 92
7.5 Squares in arithmetic progression......Page 94
7.6 One square twice another?......Page 97
7.7 Postscript on proofs......Page 100
Exercises......Page 104
8.1 Cumulative sums......Page 106
8.2 Last digit......Page 108
8.3 Coin changing with cubes......Page 110
8.4 More sums of cubes......Page 112
8.5 One cube twice another?......Page 115
Exercises......Page 117
9.1 Details of the procedure......Page 119
9.1.1 The case k = 2......Page 120
9.1.4 The case k = 5......Page 121
10 Powers of 2 and 3......Page 123
10.1 Last digit......Page 124
10.2 Number of digits......Page 128
10.3 First digit......Page 131
10.4 Coins and weights......Page 133
10.5 Repeated division......Page 134
10.6 The Tower of Hanoi......Page 137
10.7 Mixing the powers of 2 and 3......Page 138
10.8 Sums of consecutive integers......Page 141
Exercises......Page 143
11.1 Infinitude of the primes......Page 145
11.2 Mersenne and Fermat primes......Page 155
11.3 The Euclidean numbers......Page 161
11.4 Unsolved problems......Page 162
Exercises......Page 164
12.1 Triangular numbers......Page 166
12.2 Pentagonal numbers......Page 173
Exercises......Page 177
13 Egyptian Fractions......Page 179
13.1 Divergence of the harmonic series......Page 180
13.2 An Egyptian algorithm......Page 187
Exercises......Page 190
14 Square Roots......Page 192
14.1 Closeness......Page 193
14.2 Sums of square roots......Page 197
14.3 A square root spiral......Page 202
14.4 A random sequence?......Page 203
Exercises......Page 205
15.1 Family planning......Page 206
15.2 A formula for Fibn......Page 208
15.3 Properties of the Fib numbers......Page 211
15.4 The golden ratio......Page 213
15.5 Another formula for Fibn......Page 219
Exercises......Page 221
1 The method......Page 223
2 Solved examples......Page 225
Exercises......Page 226
Appendix B: Solutions......Page 227
Appendix C: List of References......Page 249