9.3 Geometric Sequences

• In a Geometric sequence, each term is found by multiplying the previous term by a constant, and the it is common ratio(r). 

          Example:  1, 2, 4, 8.....
                          The sequence starts at 1 and next term will doubled for each time
                           so we would know that r =2/1=2
          Also, when r=0, then it is not geometric.

• The nth term of a geometric sequence

           a_n = a₁ r^{n-1
           Example: 10, 30, 90, 270....
                           the 4th term would be: a}₄ = 10×3^(4-1) = 10×3³ = 10×27 = 270

•  The sum of a finite geometric sequence

• The sum of an infinite geometric sequence

Pythagorean theorem

Greek philosopher and mathematician Pythagoras lived around the year 500 BC and is known for his Pythagorean theorem relating to the three sides of a right angle triangle: a² + b² = c². Here is a way to prove it. 

 

To find the area of this trapezoid, S=(1/2)(a+b)(a+b)=(1/2)(a+b)² and you can also add the three triangles together. S=(2)(1/2)ab+(1/2)c²

Set the to equations equal, and you will get a²+b²=c²

Well-ordering priciple

The well-ordering principle is a concept, which is equivalent to mathematical induction. The theorem stated that every non-empty subset of the natural numbers has a least element.

Proof: Let A be a non-empty subset of N. We wish to show that A has a least element, that is, that there is an element a∈A such that a is greater or equal to n for n∈A.

                            P(n) : “If n ∈ A, then A has a least element.

Basic Step: P(0) is clearly true, since 0 ≤ n for all n ∈ N.

9.2 Arithmetic Sequences

• A sequence whose consecutive terms have a coman difference is called an aeuthmetic sequence. 

       Example: 1, 4, 7, 10, 13, 16, 19, 22, 25....

                      This sequence has a difference of 3 between each number.

• The nth term of an arithmetic sequences 

          The nth term of an arithmetic sequence has the form: a_n = a₁ + d(n-1)
         a_{1= the first term, d=common difference
              Example: find the 4th term by using the} a_n formula 3, 8, 13, 18, 23, 28, 33,38...
          a₄ = a₁ + d(n-1)
            = 3+5(4-1)
               = 18

• The sum of an arithmetic sequence

             

9.1 Sequences and Summation Notation

Today, we have the lesson of sequences, which is we learned in Algebra 2. A sequence is a function whose domain is the natural numbers. Rather than using function notation, however, sequence are usually written using the  a_n notation. There are infinite sequences whose domain is the set of all positive ingers, and if the domain is the set of the first n postive integers we called finite sequences.

Interesting math facts

Hey guys, I found some interesting fun facts. Hope you guys enjoy it. 

1 out of 350,000 Americans get electrocuted in their life.

1 in 8 Americans has worked at a McDonalds restaurant.

10,000 Dutch cows pass through the Amsterdam airport each year.

7 out of 10 people believe in life after death.

Google Site

https://sites.google.com/a/maranathastudents.org/chapter-8-test-review/

Cramer's Rule VoiceThread

Cramer's Rule

8.4 The Determinant of a Square Matrix

http://www.educreations.com/lesson/view/36/16765257/?s=EBXps5&ref=appemail

Cryptography

8.2 Matrices and systems of equations

A system of equation with no solution which happens when we solve the system and end up an equation that cannot make any sense mathematically. 
Example 1:

From above, we can notice that it is impossible for 0=4, thus, we can say that the system of equations has NO SOLUTION or inconsistent.

Also, we learned another type of system of equations which is a system with an infinite number of solutions. It always end up with an equation where all the variables are disappeared.

Example 2:

In the last row of matrix, it end up with all zeros on both sides of equation, which means the system of equations have an infinite set of solutions. 

8.3 The inverse of a sequare matrix

The first one would not be invertible which means there does not exist the inverse of matrix, since the third equation is all zeros. Singular. 

Also, we can use matrix multiplication to check the result.