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

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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