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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
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Numbers

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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
MelikyanElem Numb Theory/Fall05
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Sequences

A sequence {an} is a list of elements a1, a2, a3, . . .

A Geometric Progression is a sequence of the form
a, ar, ar2, ar3, …ark ,. . .
an = 3(7)n , a1 = 21, a2 =147 what about a23 ?
Triangular numbers t1, t2, t3, . . . tk, . . . Is the sequence where tk 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
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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.
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Sums and products, notations and properties.
MelikyanElem Numb Theory/Fall05
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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. Let Sn denote the sum of the first n natural numbers, that is
Sn = 1 + 2 + 3 + … + n. Prove that Sn = n(n+1)/2
First base case: n =1
Second Induction Hypothesis:
Example2: Prove that 2n > n for all positive integers
MelikyanElem Numb Theory/Fall05
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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
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The Fibonacci Numbers
Definition: The Fibonacci sequence is defined recursively by
f1 = 1, f2 = 1 and fn = fn-1 + fn – 2
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .
What is the sum of first Fibonacci numbers?
f1 + f2 + f3 + … + fn = fn+2 - 1
What about the following sum
f1 + f3 + f5 + … + f2n -1 = f2n
MelikyanElem Numb Theory/Fall05
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More examples:
Let assume that α and β (α > β) and is the root of the
following quadratic equation
x2 – x -1 =0
Use mathematical induction to verify that
fn = (αn - βn)/(α - β)
MelikyanElem Numb Theory/Fall05
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Binomial Coefficients
How to compute
(x + y)n =
(x + y)2 =
(x + y)3 =
(x + y)4 =
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