Arithmetic & Geometric progression

Arithmetic&Geometric progression

 In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2.

If the initial term of an arithmetic progression is ${\displaystyle a_{1}}$ and the common difference of successive members is d, then the nth term of the sequence (${\displaystyle a_{n}}$) is given by:

${\displaystyle \ a_{n}=a_{1}+(n-1)d,}$

and in general

${\displaystyle \ a_{n}=a_{m}+(n-m)d.}$

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

The behavior of the arithmetic progression depends on the common difference d. If the common difference is:

• Positive, then the members (terms) will grow towards positive infinity.
• Negative, then the members (terms) will grow towards negative infinity.

geometric progression

n.

A sequence, such as the numbers 1, 3, 9, 27, 81, in which each term is multiplied by the same factor in order to obtain the following term. Also called geometric sequence.

A sequence of numbers in which each number is multiplied by the same factor to obtain the next number in the sequence; a sequence in which the ratio of any two adjacent numbers is the same. An example is 5, 25, 125, 625, ..., where each number is multiplied by 5 to obtain the following number, and the ratio of any number to the next number is always 1 to 5. Compare arithmetic progression.

Examples

https://www.youtube.com/watch?v=gua96ju_FBk

References:

http://www.thefreedictionary.com/geometric+progression

Linear programming

Linear Programming

What is a linear programming?

Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, and then determining the "best" production levels for maximal profits under those conditions.

 • Optimization problem consisting in

 • maximizing (or minimizing) a linear objective function 

 • of n decision variables • subject to a set of constraints expressed by linear equations or inequalities.  • Originally, military context: "programme"="resource planning". Now "programme"="problem" 

Examples:

The three inequalities in the curly braces are the constraints. The area of the plane that they mark off will be the feasibility region. The formula "z = 3x + 4y" is the optimization equation. I need to find the(x, y) corner points of the feasibility region that return the largest and smallest values of z.
My first step is to solve each inequality for the more-easily graphed equivalent forms:

                    click this

It's easy to graph the system: 

So the corner points are (2, 6), (6, 4), and (–1, –3).
Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. So, to find the solution to this exercise, I only need to plug these three points into "z = 3x + 4y".

(2, 6):      z = 3(2)   + 4(6)   =   6 + 24 =   30 
(6, 4):      z = 3(6)   + 4(4)   = 18 + 16 =   34 
(–1, –3):  z = 3(–1) + 4(–3) = –3 – 12 = –15


Then the maximum of z = 34 occurs at (6, 4),
and the minimum of z = –15 occurs at (–1, –3).

https://www.youtube.com/watch?v=-32jcGMpD2Q

References:

http://www-sop.inria.fr/members/Frederic.Giroire/teaching/ubinet/pdfs/slides-linear-programming-introduction.pdf

http://www.purplemath.com/modules/linprog.htm

Inequality

Inequalities

Definition:

An inequality says that two values are not equal.

a ≠ b says that a is not equal to b

There are other special symbols that show in what way things are not equal.

a < b says that a is less than b
a > b says that a is greater than b
(those two are known as strict inequality)

a ≤ b means that a is less than or equal to b
a ≥ b means that a is greater than or equal to b.

Examples:

    https://youtu.be/y7QLay8wrW8

   

  

https://www.youtube.com/watch?v=0X-bMeIN53I

References:

http://www.mathsisfun.com/definitions/inequality.html

http://www.math.com/school/subject2/lessons/S2U3L5GL.html

Image

Logarithms

Logarithms

Definition:

1. log_a x = N means that a^N = x.
2. log x means log₁₀ x. All log_a rules apply for log. When a logarithm is written without a base it means common logarithm.
3. ln x means log_e x, where e is about 2.718. All log_a rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
    Note: ln x is sometimes written Ln x or LN x.

Common Logarithm

The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 10² = 100.

Natural Logarithm

The logarithms base e of a number. That is, the power of e necessary to equal a given number. The natural logarithm of x is written ln x. For example, ln 8 is 2.0794415... since e^2.0794415... = 8.


Rules
1. Inverse properties:   log_a a^x = x   and   a^(loga ^x) = x
2. Product:  log_a (xy) = log_a x + log_a y

Examples:

Write each of the following in logarithmic form:

Evaluate the following:

 What number is n?

                               

https://www.youtube.com/watch?v=mQTWzLpCcW0

click the link below for more examples!

References:

http://www.themathpage.com/aprecalc/logarithms.htm#ex4

Indices

Indices & the Law of Indices

Introduction

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.

what's are indices?

Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3⁴ and 3² can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 3⁵ and 5⁷ as their base differs (their bases are 3 and 5, respectively).

Six rules of the Law of Indices

Rule 1: 


 

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References  

http://mathematics.laerd.com/maths/indices-intro.php