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Number

Notes

Natural numbers are the set of numbers used for counting and ordering. They begin from 0 or 1, depending on the context. When 0 is included, the set is typically written as: {0,1,2,3,4,5,… }
Whole  numbers  Definition: Whole numbers are the set of numbers that start from 0 and go up to infinity in increments of 1. Set Notation: They can be represented as {0,1,2,3,4,…}
Integers are the set of numbers that include all positive whole numbers, negative whole numbers, and zero. Set Notation: They can be represented as {…,−3,−2,−1,0,1,2,3,…} Integers therefore are an extension of natural numbers. Every natural number is an integer.  

Reciprocal

 

Factors
Definition: Factors of a number are integers that divide that number exactly without leaving a remainder.
Example: For the number 12, the factors are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 exactly.

Prime Numbers

Definition: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, it can only be divided exactly by 1 and the number itself.

Examples: The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. Note that 2 is the only even prime number.

Prime Factors

Definition: Prime factors are the prime numbers that multiply together to give the original number. Every integer greater than 1 can be uniquely factored into prime numbers.

  • Example: To find the prime factors of 30, you would break it down into prime numbers:
  • 30 can be divided by 2 (the smallest prime number), giving 15.
  • 15 can be divided by 3 (the next smallest prime number), giving 5.
  • 5 is a prime number itself.
  • So, the prime factors of 30 are 2×3×5.

Prime factor tree

Prime Factor Tree
30
2
15
3
5

Multiples

Definition: Multiples of a number are the results of multiplying that number by the integers (1, 2, 3, …). In other words, a multiple of a number is any number that can be evenly divided by that number.

Example: The multiples of 4 are 4, 8, 12, 16, 20, and so on. These are obtained by multiplying 4 by 1, 2, 3, 4, 5, and so forth.

Practice

Definition

A quadratic sequence is defined by the recurrence relation involving a quadratic polynomial. For a sequence where the

nn

-th term is given by

an=an2+bn+ca_n = an^2 + bn + c

, the difference between consecutive terms changes linearly.

Characteristics

  1. Second Differences: The sequence has a constant second difference. This is the difference between consecutive differences of the terms in the sequence.

  2. Quadratic Formula: Each term

    ana_n

    of the sequence can be written as

    an2+bn+can^2 + bn + c

    , where

    aa

    ,

    bb

    , and

    cc

    are constants.

Finding the Terms

  1. Identify the Terms: Given a sequence, find the first few terms.
  2. Calculate First Differences: Find the differences between consecutive terms.
  3. Calculate Second Differences: Find the differences between consecutive first differences.
  4. Check Constancy: If the second differences are constant, the sequence is quadratic.

Example

Consider the quadratic sequence: 2, 6, 12, 20, 30.

  1. Calculate First Differences:

    • 6 – 2 = 4
    • 12 – 6 = 6
    • 20 – 12 = 8
    • 30 – 20 = 10

    First differences: 4, 6, 8, 10

  2. Calculate Second Differences:

    • 6 – 4 = 2
    • 8 – 6 = 2
    • 10 – 8 = 2

    Second differences: 2, 2, 2

    The second differences are constant, so the sequence is quadratic.

  3. Determine the Quadratic Formula: To find the exact formula, solve the system of equations based on the first few terms and their positions. For this sequence, you’d solve:Solving these equations, we get 

    a=1a = 1

    ,

    b=1b = 1

    , and

    c=0c = 0

    . Therefore, the formula for the sequence is

    an=n2+na_n = n^2 + n


Quadratic Sequence

Quadratic Sequence Example

Sequence: 2, 6, 12, 20, 30

Sequence Table

n Term First Difference Second Difference
1 2 - -
2 6 4 -
3 12 6 2
4 20 8 2
5 30 10 2

Quadratic Formula

The quadratic sequence formula is given by:

an = n2 + n

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