Alg End Term Exam Reviewer

8/8/2019 Alg End Term Exam Reviewer

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End Term Exam Reviewer

2nd

Term: Algebra

- The Great Codigo 2.0

o Arithmetic Sequences

d = A

n A

n-1

An = A1 + (n-1)d

Sn = (n/2)(A1 + An)

Sn = (n/2)(2A1 + (n-1)d)

o Geometric Sequences

R = An/ An-1

An = A1Rn-1

Sn = A1 / ( 1 r)

Sn = (A1(1 - Rn)) / 1 R

o

- Patterns

o Anything with a consistent arrangement

o Ex.

1,2,3,4,5,

1,3,7,13,21

Up,up,down,down

Left,right,left,right

- Sequences

o A set of numbers arranged in a definite order

o Common difference an amount added to a part in a sequence to get the next part

o As long as there is a pattern, its considered a sequence

o 2 types

Arithmetic adding

Geometric multiplication

o Ex:

1,2,3,4,5

1,3,7,13,22

1,2,4,8,16

o Harmonic Sequences

Just reciprocate an arithmetic sequence

1 , 2 ,3 = 1/1, ½, 1/3

o Recursive formula

Using the values of the first 3 terms, you can find a formula to find the next terms

Given

y 3, 9, 27, 81, 243

y A1=3, A2=9, A3=27

y If A2 = A1 X 3; and A3 = A2 X 3 ; R = 3

y Then An = RAn-1

o Sigma

Used for notation of summations

5

[x2]

x=1

What that means is that you take x2

for x=1, evaluate it (1), and then do the same for x=2,



x=3 ... x=5 and then add each result. So in this case, the above summation would

evaluate to 55 (12+ 2

2+ 3

2+ 4

2+ 5

2= 55).

- Series

o The sum of all the terms of a sequence

o Two types

Finite the sequence has an end

Infinite unending sequence

- Matrices

o Numbers arranged in a rectangular manner

o Dimensions are read: (Rows) x ( Columns)

Matrices can be named using letters

o Numbers in a matrix are called elements

Multiply the dimensions of a matrix, and you get the number of elements it has

A element can be named A12 : you find it on the 1st

row, on the 2nd

column

o 4 types

Row 1 row

Column 1 column

Square equal # of columns and rows

Zero all elements are 0so Matrices are equal when they have the same dimensions, and each element is equal in value to its

corresponding element in the opposite matrix

- Matrix addition and subtraction

o Only matrices with equal dimensions can be added/subtracted

o Simply add/ subtract elements by their corresponding element

o Inverses

When asked for the inverse, reverse the polarities of your elements

y Ex: [ -1 2] would inversely be [1 -2]

o Properties of addition/subtraction

Associative

y (A+B)+C = A + (B + C)

Commutativey A + B = B + A

Distributive

y s(A+B) = sA + sB

- Matrix Multiplication

o Multiplying matrices (duh)

o You can only multiply matrices when:

The Columns of the first matrix is equal to the row of the second

(3 X 2) X (2 X 4) = ok

(2 X 4) X (3 X 2) = hell no

o The resulting matrix will have rows equivalent to the first matrix and columns equivalent to the second

(3 X 2) X (2 X 4) = (3 X 4)

o HOW TO MULTIPLY Given A x B

Multiply the terms of Matrix As first row with matrix Bs first Column

y Add the resulting numbers

Multiply the terms of Matrix As first row with Column Bs next column

y Add the resulting numbers

Rinse and repeat till you finish off all the columns

Do the same with Matrix As other rows after finishing the first



When putting the finished numbers on your new matrix, fill the elements from left to right, up

going down; like reading a book

- Determinants

o A number assigned to a matrix, determined by adding the products of certain numbers

o Given a 2 x 2 matrix

Simply cross multiply, then subtract

o Given a 3 x 3

- Cramers Rule

o A process which solves linear equations

o 2 x 2

Linear System Coefficient Matrix

ax+by=e

cx+dy=f

Det A = ad - bc

o 3 x3

First, make a 3 x 3 matrix using the coefficients, just like 2x2

Get the determinant

When solving for the value of a variable, replace the column of coefficients belonging to that

variable with the constants on the right side of your equations

y Divide the result by your determinant

Rinse and repeat with the other variables

NOTE: If a variable doesnt exist in one of the 3 equations, add one:

y Ex: 3y + 4z = 5; 0x + 3y +4z = 5

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Alg End Term Exam Reviewer