10.1: Infinite Sequences and Summation Notaion

A sequence is a function whose domain is all natural numbers.

 represents the first time of the sequence, while  denotes a positive integer, or the nth term.
Sequences often appear as  , representing the first three terms of the sequence, as well as the nth term.

An infinite sequence is a function whose domain is the set of positive integers.

For example: Find the eighth term of the sequence,  with the first four terms: 1/3, 1/2, 3/5, 2/3...
Eighth term:
 n=8
= 4/5

There are two basic forms of writing sequences, recursive  and explicit.


Recursive Form






On the other hand, the same sequence may be represented in this way as well:

Explicit Form




Summation Notation
Summation notation is another alternative form of a sequence that can be used to find the sum of multiple terms of an infinite sequence.

Summation Notation
 

k represents the lower limit, while m represents the upper limit.
k also represents the index of summation.

This picture also re-explains summation notation in slightly different terms.


There are also two rules that apply to the sums of a constant.

Theorems on the Sum of a Constant:

1) (c being a constant)


2)


There are also a few theorems that derive from the following original theorem on sums.

Theorem on Sums


Addition

Subtraction


Constants
 for every real number c


To evaluate a sum:

Find the sum:



To find the sum, substitute the integers 1 (lower limit) through 3 (upper limit) in for k and add the resulting terms together.

(1+1) + (2+1) + (3+1)= 9

That's it for now, thank you!

Julia Wilkins

9.5 - Systems of Linear Equations in More Than Two Variables (Matrices)

A Matrix is a rectangular array of numbers, symbols, or expressions. 


The rows of a matrix are the numbers that appear horizontally, and the columns of a matrix are the numbers that appear vertically next to each other.

The matrix that is obtained from a system of linear equations is called the matrix of a system.

For example:  

turns into the matrix of:

 

where the numbers in the matrix (elements) are the coefficients of the system of linear equations in the previous set of equations.  These kinds of matrices are called augmented matrices because the matrix of the system can be obtained from the coefficient matrix.

Theorem on Matrix Row Transformations:

          Given a matrix of a system of linear equations, a matrix of an equivalent system results if :

                 1) two rows are interchanged

                 2) a row is multiplied or divided by a non zero constant

                 3) a constant multiple of one row is added to another row

Using matrices to solve a system of linear equations 

turns into the matrix:

Now, once you have 0's under both of the first numbers for the 2nd and 3rd rows, you go to the second row and multiply it by the number that will make the 1st non-zero number equal to 1.  In this case you multiply the row by 1/5 to make 5 equal to 1.

This is called the echelon form of a matrix.  From this, you get that:

1z=2 so z=2

and you can plug that into the original equations to solve for x and y using substitution.  

Another way to solve a matrix problem is to plug the original coefficient matrix into a calculator and go to the math matrix menu and get the reduced row echelon form of the matrix, giving you the 3 solutions. (x,y,z)

The solutions found to this equation after going back and solving by substitution are x=4, y=3, and z=2

Echelon form of a matrix:

  1) the first nonzero number in each row is 1

  2) the column containing the first nonzero number in any row is to the left of the column containing the first nonzero number in the column below

  3) rows consisting entirely of 0's may appear at the bottom of the matrix, but no where else

Guidelines for finding the echelon form of a matrix:

  1) locate the 1st column that contains nonzero numbers, and apply transformations so that the number 1 is in the first row of that column

 

  2) apply row transformations of the type  above for j>1 to get 0 under the number 1 obtained in step 1 for each remaining row.

   

  3) disregard the first row. Locate the next column with nonzero elements, and apply transformations to get the number 1 into the second row of that column.

 4) apply row transformations of the type above for j>2 to get 0 under the number 1 obtained in step 3 of this process for each remaining row.

5) disregard the first and second rows.  Locate the next column that has nonzero elements and repeat the procedure again.

6) continue this process until the echelon form is completed.

That's about all you need to know about matrices, although the easiest way to do them is just by plugging the matrix into a calculator to find the reduced row echelon form.  Here are a couple useful links in helping to solve matrices.

http://www.wyzant.com/Help/Math/Precalculus/Matrices/Reduction_to_Row_Echelon_Form.aspx

http://www.youtube.com/watch?v=pyUEsefV4uY

Thanks, 

Michael Levitsky