Numbers, Sequences and Sums

In this lecture we introduce basic materials that will Frequently be used throughout the course.

We will cover the important sets of numbers , the concept of integer sequences, summations and products.

Instructor: Hayk Melikyan [email protected]

MelikyanElem Numb Theory/Fall05 1

Numbers

    Integers.

The Well-Ordering Property:

Every nonempty set of positive integers has a least element Rationals: Irrationals: Theorem:

2

is irrational.

Algebraic numbers: A number which is the root of polynomial with integer coefficients. 5, ¾, or

2 •Transcendent numbers: e, π, 3 2 MelikyanElem Numb Theory/Fall05 2

Sequences

 A sequence {a n } is a list of elements a 1 , a 2 , a 3 , . . .

 A Geometric Progression is a sequence of the form a, ar, ar 2 , ar 3 , …ar k ,. . .

a n = 3(7) n , a 1 = 21, a 2 =147 what about a 23 ?

Triangular numbers t 1 , t 2 , t 3 , . . . t k , . . . Is the sequence where t k is the number of dots in the triangular array of k rows with j dots in jth row.

 An Arithmetic Progression is a sequence of the form a, a+d, a+2d, . . . a + (n-1)d, . . . MelikyanElem Numb Theory/Fall05 3



Countable sets

A set is called

countable

if it is finite or there exists a one –to –one correspondence between the set of positive integers and the set.

Theorem : The set of rational numbers are countable. 

Sums and products, notations and properties.

MelikyanElem Numb Theory/Fall05 4

Mathematical induction

Theorem:

A set of positive integers that contains the integer 1, and has the property that, for every positive integer n, if it contains all positive integers 1, 2, 3, …, n, then

it also contains the integer n+1, must be the set of all positive integers.

Examples: 1.

S n Let Sn denote the sum of the first n natural numbers, that is = 1 + 2 + 3 + … + n. Prove that S n = n(n+1)/2 First base case: n =1 Second Induction Hypothesis: Example2: Prove that 2 n > n for all positive integers MelikyanElem Numb Theory/Fall05 5

Recursive definitions

We say that function f(n) is drfined recursivly if The value of f (1) is specified and for each positive integer n the rule is given for coumpiting f(n+1) from f(n).

Example: factorial function MelikyanElem Numb Theory/Fall05 6

The Fibonacci Numbers

Definition: The Fibonacci sequence is defined recursively by f 1 = 1, f 2 = 1 and f n = f n-1 + f n – 2 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .

What is the sum of first Fibonacci numbers?

f 1 + f 2 + f 3 + … + f n = f n+2 - 1 What about the following sum f 1 + f 3 + f 5 + … + f 2n -1 = f 2n MelikyanElem Numb Theory/Fall05 7

More examples: Let assume that α and β (α > β) and is the root of the following quadratic equation x 2 – x -1 =0 Use mathematical induction to verify that f n = (α n - β n )/(α - β) MelikyanElem Numb Theory/Fall05 8

Binomial Coefficients

How to compute (x + y) n = (x + y) 2 = (x + y) 3 = (x + y) 4 = MelikyanElem Numb Theory/Fall05 9