9.1 Sequences and Summation Notation

A few of the things we learned to do in this lesson are:
1. Find terms of a sequence.
2. Write equations of sequences.
3. Evaluate factorials.

1. Sequences are often represented as an equation like an = 2n - 1. To find a specific term, like the third term (a3), plug the number of the term into the equation as n. 2(3) - 1 = 5.

2. Finding the equation of a sequence can be tricky, but the best method is to just list the sequence with the number of each term under it., like so:                                  sequence:     1  3  5  7
                                                                                                                n:     1  2  3  4
The pattern can be determined from there.

3. A factorial (!) is a product of consecutive natural numbers, leading up to the number by which the product is represented.
For example, 5! = 5x4x3x2x1.

A lot of times equations that involve factorials can appear overwhelming at first, but simplification is possible if there is division involved:

Example:   100!/99! -----> The denominator cancels out the first 99 factors of the product 100!, so the only number left is 100. Therefore, the answer is 100.

Most of these problems involve a lot of logical thinking rather than plug-and-chug equations. This is good for some students who understand that logic, but it can be a struggle for other students.

Link to the Google Site for Chapter 8 Review

https://sites.google.com/site/chapter8reviewgroup3/

Educreations: Determinant of a Matrix

http://www.educreations.com/lesson/view/determinant-of-a-matrix/

VoiceThread: Testing Collinear Points

Cryptography

On Wednesday, we learned about a form of cryptography, or code language, that uses matrices. In the first step of this worksheet, we had to use an encryption matrix and code Miss V gave us to find out the meaning of the code. We had to use the inverse of the encryption matrix, which is its decryption matrix, to decode the message into new numbers (3 numbers at a time to multiply by the 3X3 decryption matrix). Finally, we had to transfer those new numbers into letters (A=1, B=2, etc.), and the code spelled out the words "MEET ME MONDAY". I then created an encryption matrix of my own in step 2 and made a code. Can you figure out what the code means?

Recognizing Type of Solutions to Systems of Equations

Solving systems of equations using matrices is very helpful, as there are several tricks to be utilized to simplify the system quickly. The most common use of the Elementary Row Operations are to simplify equations into row echelon form.

Row Echelon Form Requirements:
1. All rows consisting entirely of zeros occur at the bottom of the matrix
2. For each row that does not consist entirely of zeros, the first nonzero entry is a 1.
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

From row echelon form, we can find the type of solutions the system will have. In the following example, we can tell that the system will have infinitely many solutions from the row echelon form.

(b) 

                                        

                                  

 Since the bottom row would transfer into the equation 0 = 0, the other equation x + (5/2)y = -1/2 is the only relevant equation. Therefore, any number x will result in a number for y, so there are infinitely many solutions.

Here's another example:

            

                                                        

                                     

                              

If you switch the final two equations at the end of this example, it is in row echelon form. That final line would transfer into the equation 0+0+0=12. Obviously, this equation is incorrect, so the system as a whole is inconsistent and therefore has no solutions.

Sources:
http://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx

Inverse Matrices

In both of these examples, I'm attempting to find the inverse of the original given matrix. The idea is that a matrix times it's inverse will equal the identity matrix. The identity matrix is a matrix with all zeroes, but a main diagonal of all 1s. To complete these problems, write the matrix along with an identity matrix in the same box. Then, simplify until the only nonzero numbers are ones on the main diagonal, just like you would do with a normal simplifying matrices problem. The first of these examples didn't work because there was a row of zeroes, so it couldn't be out in the proper form of only ones on the main diagonal. This means that the system is singular, not inversible, and does not have an inverse.