Pascal's Triangle

One can use the Binomial Theorem or the Pascal Triangle to expand and equation such as (2x+3y)^4 without just multiplying it all out manually! However, as Miss V quietly states, Pascal's Triangle is the one that a person should use. With the Pascal's triangle method, you find the row according to the power of your equation. Then, you fill in the number after each constant starting with the highest power of the first term and descending and the lowest number of the second term and ascending after each term in your expansion. With the use of the Pascal's Triangle, one can also find the term that Combinations are trying to find by looking at the first number before the C for the row number and the number after the C for the term. However, you must subtract 1 to the term number to get the correct answer.
Here is a Pascal Triangle: 

Mathematical Induction

In Mathland, we learned about how we can prove a statement using the platform of Mathematical induction. Mathematical induction is very simple, but is long and has clear steps and things you must write every time for it to hold true! For example, without the statement: "assume the statement is true for n" at the start of step 2, it would not hold true! With mathematical induction, you must memorize the format in order to do it. Mainly, there are two steps. The first step is prove the statement is true at the starting point while step 2 is to assume the statement is true for n and then prove that the statement is true for n+1.
Here is a picture of the format you must follow, the orange parts hold three for all mathematical induction proofs

Well Ordering Principle

Basically, the Well Ordering Principle or Well-ordering theorem is a term in Math that within a set of positive integers, there are is a least element, which is self-explanatory for it is the number that is the least in value. this principle turns out to be logically equivalent to the mathematical induction, the fifth axiom of Peano, which is quite surprising. If mathematical induction is true for all natural numbers, then so is the well-ordering principle. Basically, the well-ordering principle can be proved with the mathematical induction. Ultimately, both imply each other to work because they are based on the same type of logic.
Here is an the proof of well-ordering principle according to mathematical induction: the http://www.math.wustl.edu/~chi/310notesIV.pdf

0!

Most factorials are easy to understand and to find the solution to, except the factorial of a certain number. This number is 0. It is weird that every other number above zero is basically that number multiplied by all numbers below it towards one. However, 0 x 0 should be zero for even if it is multiplied by one, it is still zero. However, through this article, one can basically understand why mathematicians had defined 0! As 1. One huge reason is due to permutations. A set with no values should be arranged in only one way. Also, the formulas for permutations and combinations need the value of zero factorial to be one so that it is consistent with the equations that are derived. It is basically made up so that math equations stay true. One can learn more from this article: http://statistics.about.com/od/ProbHelpandTutorials/a/Why-Does-Zero-Factorial-Equal-One.htm

Arithmetic Sequences

We learned about arithemetic sequences. Arithmetic sequences are sequences of numbers where there is a common difference between each number such that a number is is being added to each number to get the next number in the sequence. There are both explicit and sum formulas. Explicit formulas are used to find a single number in a sequence when the first number and the change or common difference between each number is known. The sum formulas are used to find the sum of the whole arithmetic sequence when the first and last number and the amount of numbers are known. The equations and examples will be in a picture below. 

Sequences and Summation

In Mathland, we learned about sequences and summation. A sequence can be infinite and or finite and that appellation pretty much sums up how they are. Numbers with an exclamation mark can also be known as factorials. A factorial is basically the number that is being factorized times all the integers lower than it to zero. For example: 4!=4x3x2x1. However, the factorial of zero is one due to some important reasons. Summation notation is a way one can sum up sequences when the upper limit, explicit formula, and lower limit are known. There are also many properties of summation notations. Below are summations notations and their properties.

Determinants

This was my group's educreation video on how to use determinants:
http://www.educreations.com/lesson/view/p-635-36/16782864/?s=RdR6UY&ref=app