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

250 Pages·2019·2.18 MB·English

by  Shailesh Shirali

Mathematical Marvels
A Primer on
NUMBER SEQUENCES
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Mathematical Marvels
A Primer on
NUMBER SEQUENCES
Shailesh Shirali
Universities Press (India) Private Limited
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© Universities Press (India) Private Limited 2001
First published 2001
Reprinted 2004, 2010, 2014
eISBN 978-81-7371-967-7
e-edition:First Published 2015
ePUB Conversion: Techastra Solutions Pvt. Ltd.
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Contents
Title Page
Copyright
Preface
Part A The Method of Differences
1 Introduction
1.1 What is a pattern?
1.2 Number sequences
1.3 Sequences in mathematics
2 Terms and Symbols
3 Polynomial Sequences
3.1 Introduction
Exercises
3.2 A symbolic apparatus
Exercises
4 The Generating Formula
4.1 Introduction
4.2 A worked example
4.3 Warning!
4.4 Interlude: Hypotheses in science
Exercises
4.5 Triangular numbers
4.6 Diagonals of a polygon
4.7 Pentagonal numbers
4.8 Sums of squares
4.9 Proof of the formula
Exercises
5 Exponential Sequences
5.1 Sequences of the form {an}
5.1.1 The Sequence {(- 1)n}
5.1.2 The Sequence {2n}
5.1.3 The Sequence {3n}
5.2 Mixed sequences
Exercises
5.3 Seeking an explanation
Exercises
6 Conclusion
6.1 General comments
Exercises
List of references
Part B A Gallery of Sequences Introduction
7 The Sequence of Squares
7.1 Generating the squares
7.2 Last digit
7.3 Coin changing using squares
7.4 Sums of squares
7.5 Squares in arithmetic progression
7.6 One square twice another?
7.7 Postscript on proofs
Exercises
8 The Sequence of Cubes
8.1 Cumulative sums
8.2 Last digit
8.3 Coin changing with cubes
8.4 More sums of cubes
8.5 One cube twice another?
Exercises
9 A Curious Procedure
9.1 Details of the procedure
Examples
9.1.1 The case k = 2
9.1.2 The case k = 3
9.1.3 The case k = 4
9.1.4 The case k = 5
10 Powers of 2 and 3
10.1 Last digit
10.2 Number of digits
10.3 First digit
10.4 Coins and weights
10.5 Repeated division
10.6 The Tower of Hanoi
10.7 Mixing the powers of 2 and 3
10.8 Sums of consecutive integers
Exercises
11 The Sequence of Primes
11.1 Infinitude of the primes
11.2 Mersenne and Fermat primes
11.3 The Euclidean numbers
11.4 Unsolved problems
Exercises
12 Polygonal Numbers
12.1 Triangular numbers
12.2 Pentagonal numbers
Exercises
13 Egyptian Fractions
13.1 Divergence of the harmonic series
13.2 An Egyptian algorithm
Exercises
14 Square Roots
14.1 Closeness
14.2 Sums of square roots
14.3 A square root spiral
14.4 A random sequence?
Exercises
15 Fibonacci Numbers
15.1 Family planning
15.2 A formula for Fib
n
15.3 Properties of the Fib numbers