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comp_bos · 2005 March 4th, 07:53 PM

Arithmetic progression: Its a sequence of numbers, in which difference of any two successive numbers is constant called common diference.

First term of the sequence is usually denoted by "a" and
Common difference by "d"

Example: 0, 2, 4, 6, 8, 10.....

a = 0
d =2

Tn(nth term of an AP) = a + (n-1) d
Sum of first n terms of an AP = n/2[ 2a + (n-1)d]

Geometric Progression: Its a sequence in which the ratio of any two consecutive terms is always a constant called common ratio (r).

If first term is denoted by "a" and
Common ratio by "r"

nth term of the series = ar^(n-1)
Sum of first n terms = a(1-rn)/(1-r) ; and r # 0

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Number Theory

Natural numbers (1,2,3,4,... but no zero)

Whole numbers (0,1,2,3,4,... with a zero)

Integer numbers (...,-4,-3,-2,-1,0,1,2,3,4,...)

RATIONAL NUMBERS are integers and fractions of integers put together - but there aren't any other numbers. Everything can be represented as a rational number.

IRRATIONAL NUMBERS: numbers that cannot be written as fractions of integers (like the square root of two, pi, and e)

REAL NUMBERS Rational/Irrational Put together, they make the Real numbers.

IMAGINARY NUMBERS Some numbers that don't fit into the real numbers sqrt(-1).

COMPLEX NUMBERS: REAL & Imaginary numbers put together.

complex

/ \

imaginary real

/ \

irrational rational

/ \

integers fractions

| |

whole integers

|

naturals

Integers and rational numbers.

An integer is a positive or negative whole number, or zero. A rational number is a positive or negative fraction, or zero.

Prime numbers and composite numbers.

A prime number is an integer n with no divisors other than 1 and n. A composite number is an integer which is not prime.

The first few prime numbers are 2,3,5,7,11,13,.... There are infinitely many prime numbers.

Two is the only EVEN prime number.

GCD's and relatively prime integers.

Two integers are relatively prime if they have no prime factors in common. For example, 101 and 17 are relatively prime, but 12 and 100 are not ( they are both divisible by 2).

The greatest common divisor (GCD) of two numbers a and b is the largest number dividing both a and b.

Zero: It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number, which is not zero is said to be nonzero.

Even number: is an integer of the form, where k is an integer. The even numbers are therefore, -4, -2, 0, 2, 4, 6, 8, 10…..

Set

Name

..., -2, -1, 0, 1, 2, ...

Integers

1, 2, 3, 4, ...

positive integers

0, 1, 2, 3, 4, ...

nonnegative integers

0, -1, -2, -3, -4, ...

nonpositive integers

-1, -2, -3, -4, ...

negative integers

http://mathworld.wolfram.com/topics/NumberTheory.html

Please correct me If I have something wrong here.
Please let me know If Harmonic Series is also in GMAT scope.

2005 March 4th, 07:53 PM	#1 (permalink)
comp_bos TestMagic Guru-in-Training Join Date: Nov 2004 Location: US Posts: 738	Progressions : AP , GP & Number Theory Arithmetic progression: Its a sequence of numbers, in which difference of any two successive numbers is constant called common diference. First term of the sequence is usually denoted by "a" and Common difference by "d" Example: 0, 2, 4, 6, 8, 10..... a = 0 d =2 Tn(nth term of an AP) = a + (n-1) d Sum of first n terms of an AP = n/2[ 2a + (n-1)d] Geometric Progression: Its a sequence in which the ratio of any two consecutive terms is always a constant called common ratio (r). If first term is denoted by "a" and Common ratio by "r" nth term of the series = ar^(n-1) Sum of first n terms = a(1-rn)/(1-r) ; and r # 0 ------------------------------------------------------------------------- Number Theory Natural numbers (1,2,3,4,... but no zero) Whole numbers (0,1,2,3,4,... with a zero) Integer numbers (...,-4,-3,-2,-1,0,1,2,3,4,...) RATIONAL NUMBERS are integers and fractions of integers put together - but there aren't any other numbers. Everything can be represented as a rational number. IRRATIONAL NUMBERS: numbers that cannot be written as fractions of integers (like the square root of two, pi, and e) REAL NUMBERS Rational/Irrational Put together, they make the Real numbers. IMAGINARY NUMBERS Some numbers that don't fit into the real numbers sqrt(-1). COMPLEX NUMBERS: REAL & Imaginary numbers put together. complex / \ imaginary real / \ irrational rational / \ integers fractions \| \| whole integers \| naturals Integers and rational numbers. An integer is a positive or negative whole number, or zero. A rational number is a positive or negative fraction, or zero. Prime numbers and composite numbers. A prime number is an integer n with no divisors other than 1 and n. A composite number is an integer which is not prime. The first few prime numbers are 2,3,5,7,11,13,.... There are infinitely many prime numbers. Two is the only EVEN prime number. GCD's and relatively prime integers. Two integers are relatively prime if they have no prime factors in common. For example, 101 and 17 are relatively prime, but 12 and 100 are not ( they are both divisible by 2). The greatest common divisor (GCD) of two numbers a and b is the largest number dividing both a and b. Zero: It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number, which is not zero is said to be nonzero. Even number: is an integer of the form, where k is an integer. The even numbers are therefore, -4, -2, 0, 2, 4, 6, 8, 10….. Set Name ..., -2, -1, 0, 1, 2, ... Integers 1, 2, 3, 4, ... positive integers 0, 1, 2, 3, 4, ... nonnegative integers 0, -1, -2, -3, -4, ... nonpositive integers -1, -2, -3, -4, ... negative integers http://mathworld.wolfram.com/topics/NumberTheory.html Please correct me If I have something wrong here. Please let me know If Harmonic Series is also in GMAT scope.