GeometricGeometric ADVANCED ALG/TRIG Chapter 11 – Sequences and Series Sequences (an ordered list of numbers) Geometric R = common ratio Geometric mean

Geometric

ADVANCED ALG/TRIG

Chapter 11 – Sequences and Series

Sequences(an ordered list of

numbers)

GeometricR = common

ratioGeometric mean = square root of

product of 2 numbers

Recursive formula

an = an-1 r; a1

given

Explicit formulaan = a1 r(n-1)

ArithmeticD = common

differenceArithmetic mean

= sum of 2 numbers divided

by 2 (the average)

Recursive formula

an = an-1 + d; a1

given

Explicit formulaan = a1 + (n-1)d

Series(sum of terms in a

sequence)

GeometricFINITE – ends; has a sum

INFINITEConverges when |r|

<1; approaches a limitDiverges when |r|> 1; does not approach a

limit

Sum of a Finite

Geometric Series

Sn = a1(1-rn) 1 - r

Sum of an Infinite Geometric Series

Sn = a1

1 - r

ArithmeticFinite – endsInfinite – does not end…Summation Notationuses Sigma ; has lower and upper limits

Sum of a Finite Arithmetic Series

S = n/2(a1 + an)

Conclusion about these features

Cubes are Three Dimensional

FeaturesMain ideas Features

Cube Square

Conclusion about these features

Squares areOne Dimensional

Count the sidesA cube has ______ sides.

Count the sidesA square has______ sides.

Sides

Conclusion about this main idea

The sides are shaped like

squares

Hold the block and count the cornersA cube has _______ corners.

Touch each corner with your pencilA square has______ corners.

Corners


A cube has more corners

Build something

Make a design on paper

What can you do with

it?


A square is easier to draw and use

Divide

Divide both numbers and start again

3 3-3 =1 6 6-3 =2

Is there a number that will divide into both numbers?No, the fraction is reduced 4 7Yes, keep going

Is the numerator 1 less than the denominator?Yes, the fraction is reduced 5/6

No, keep going 5/3 = 1 2/3

Is the numerator a “1”?

Yes, the fraction is reduced 1/6

No, keep going 2/4 = 1/2

How to tell if a fraction is reduced to lowest terms

Reducing Fractions Is about …

1. Identify the slope (m) and the y-intercept (b).Example: m = 2, b = -1

3. Use the slope the locate a second point.

4. Draw a line through the two points.

2. Graph the y-intercept on the y-axis. Example:

2. Find the y-intercept. Let x=0 and solve the equation for y.Example: 3(0) – 2y = 6 -2y = 6 y=-3

1. Find the x-intercept. Let y=0 and solve the equation for x.Example: 3x – 2(0) = 6 3x = 6 x=2

Slope-intercept form: y = mx + b

Example: y= 2x - 1

Standard Form:Ax + By = C

Example: 3x – 2y = 6

4. Draw a line through the two points.

3. Graph the x-intercept on the x-axis and the y-intercept on the y-axis.

Graphing Linear Equations

Key Topic

is about... Angles

4 kinds of angles

Straight Acute

Right Obtuse

180 0 180

does not bend

less than 90 0

1 -- 89 0 0

small angle

always 90 0

square corners

perpendicular

greater than 90 0

0 91 - 179 0

wide angle angle larger than right angle

type of angle is determined by the degree of arch

© 2001 Edwin S. Ellis

So what? What is important to understand about this?

SYSTEMS OF LINEAR INEQUALITIES

First Inequalit

y

1 Graph: a.Solve for y and identify the

slope and y-intercept. b.Graph the y-intercept on the

y-axis and use the slope to locate another point.

-or- find the x and y intercepts and graph them on the x and y axis.)

2 Determine if the line is solid or dashed.

3 Pick a test point and test it in the original inequality.

• If true, shade where the point is.

• If false, shade on the opposite side of the line.)

Second Inequalit

y

1 Graph: a.Solve for y and identify the

slope and y-intercept. b.Graph the y-intercept on the

y-axis and use the slope to locate another point.

-or- find the x and y intercepts

and graph them on the x and y axis.)

2 Determine if the line is solid or dashed.

3 Pick a test point and test it in the original inequality.

• If true, shade where the point is.

• If false, shade on the opposite side of the line.)

Solution

1 Darken the area where the shaded regions overlap.

2 If the regions do not overlap, there is no solution.

Check

1 Choose a point in the darkened area.

2. Test in both original inequalities.

3. Correct if both test true.

Copyright 2005 Edwin Ellis

Key Topic

is about... POLYGONS

labeling shapes according to the number of sides

Triangle Quadrilateral Pentagon Hexagon

Octagon Decagon

3 sides

3 angles

4 sides

4 angles

5 sides

5 angles

6 sides

6 angles

8 sides

8 angles

10 sides

10 angles

TRI means 3 QUAD means 4 Pent means 5 Hex means 6

Oct means 8 Dec means 10

Polygon is a closed, flat figure with straight lines for sides

© 1998 Edwin S. Ellis


These are the steps to …Ratio Method Factor Ax2 + Bx + C

Five StepsCopyright 2003 Edwin Elliswww.GraphicOrganizers.com

Write the first ratio in the first binomial and the second ratio in the second binomial. 8(3x – 1) ( x – 1) Check using FOIL.

Step 5

Write the ratio A/ Factor. Write the ratio Factor / C. Reduce. 3/1 1/1 are reduced.

Step 4

Example: 24m2 – 32m + 8

Factor out the GCF 8(3m2 – 4m + 1)

Write two binomials 8 ( ) ( )Signs: + C signs will be the same sign as the sign of b . - C negative and positive

8( - ) ( - )

Find AC

AC = 3

Find the factors of AC that will add or subtract (depends on the sign of c) to give u B. 3 and 1 are the factors of AC that will add (note c is +) to give B.

Step 1

Step 2

Step 3

Why are these steps important?

Following these steps will allow one to factor any polynomial that is not prime.

Factoring a quadratic trinomial enable one to determine the x-intercepts of the parabola.

Factoring also enables one to use the x-intercepts to graph.

menu

Key Topic is about...

Essential Details

Divisibility

two (2) - if a number ends with 0,2,4,6,8

dividing numbers that don’t have remainders

five (5) - if a number ends with 0 or 5

ten (10) - if a number ends with 0

three (3) - if the sum of the digits can be

nine (9) - if the sum of the digits can be

It’s an easier way to divide BIG numbers!

Lesson by Tuwanna McGee


divided by 3

divided by 9

Detail Essential because...

Main idea

Detail Essential because...

Main idea

is about... Lines

2 kinds of lines

ParallelPerpendicular

Intersect / meet

form right angles

forms a square corner

never intersect, meet, or touch

same plane

go in same direction

There are only two types of lines

© 1998 Edwin S. Ellis Key Topic


Word Walls

New WordDefinition Picture

Knowledge Connection

triangle3 sides 3 angles “tri” means 3

Looks like a pyramid

quadrilateral4 sides 4 angles “quad” means 4

Looks like a box

hexagon6 sided box; 6 angles “hex” means 6

Looks like a stop sign

pentagon5 sided box; 5 angle “pent” means 5

Looks like a house

Graphing a Quadratic Function by Hand Is about …

These options allow one to graph any quadratic function. A quadratic function models many of the physical, business, and area problems one see in real world situation. For example: maximize height of a projectile; maximize profit or revenue; minimize cost; Maximize/ minimize area or volume

Main Idea

Details

1Complete the square in x to write the quadratic function in the form f(x) = a(x – h)2 + k.

2Graph the function in stages using transformations.

Option 1

Main Idea

Details

1Determine the vertex ( -b/2a, f(-b/2a)).2Determine the axis of symmetry, x = _b/2a3Determine the y-intercept, f(0).4a) If the discriminant > 0, then the graph of the function has two x-intercepts, which are found by solving the equation.b) If the discriminant = 0, the vertex is the x-intercept.c) If the discriminant < 0, there are no x-intercepts.5Determine an additional point by using the y-intercept the axis of symmetry.6Plot the points and draw the graph.

menu

Option 2

menuCopyright 2003Edwin EllisGraphicorganizers.com

Product propertylog bxy = logbx + logb y

Power propertylog bN x = x logb N

Quotient propertylog bx / logb y = logbx -

logb y

PROP ERTIES

Exponential y = abx

Asymptote = x-axis, y = 0y-intercept (0,1)

For both graphs, relate to parent function and label

intercepts.

Logarithms Logb N = PAsymptote = y-axis, x = 0

x-intercept = (1,0)

GRAP HS

Exponential y = abx

b>1 = growth, 0<b<1 = decay

Common Log = base 10; log

Natural Log = base e; ln

Logarithms Logb N = PB = base, N = #, P =

power

EXPRESSIONS

Exponential y = abx

Take log of both sides.

Use properties whilesolving and simplify.

Logarithms Logb N = PWrite in exponential form.

EQUATIONS

Advanced Alg/Trig Chapter 8

EXPONENTIALS AND LOGARITHMS

Find the vertex and make a table.General form: y = │mx + b│ + c

So what? What is important to

understand about this?

The graph is always a “V”. A minus sign outside the absolute value bars cause the “V” to be flipped upside down. The m value in the equation affects the slope of the sides of the “V”.

1Graph the parent function. Its vertex is usually the origin.

2Translate h units left (if h is positive) or right (if h is negative).3Translate k units up (if k is positive) or down (if k is negative).

Example: y = -│2x - 4│+ 1Parent function: -│2x│

Translate the parent function.Parent function: y = │mx │ Translated form: y = │mx ± h│ ± k

Graphing Absolute Value Equations

1Find the vertex using (-b/m, c). 2Make a table of values.3Choose values for x to the left and to the right of the vertex. Find the corresponding values of y.4Graph the function.

Example: y =│2x - 4│+ 1 Vertex (-(-4)/2 , 1 ) = (2, 1) x y

3 │2(3) - 4│+ 1=│2│+1=3

1 │2(1) - 4│+1=│-2│+1=3

Hot Dog Gist & Details © 2003 Edwin Elliswww.GraphicOrganizers.com

Exponents

Zero Exponents

n0 = 1 -n0 = -1 (-n) 0 = 1

Details

Gist

Multiplying Like Bases

am an = am + n

Details

Gist

Quotient to a Power

(a/b)n = an/ bn

Details

Gist

Dividing Like Bases

am = am-n

an

Details

Gist

Negative Exponents

n-1 = 1/n 1/n-1 = nDetails

Gist

Power to a Power

(am)n = amn

Details

Gist

Product to a Power

(ab)n = an bn

Details

Gist

Details

Gist

PolynomialsAlgebraic expressions of problems when the task is to determine the value of one unknown number… X


The concept is important for the AHSGE Objective I-2. You may add or subtract polynomials when determining or representing a customer’s order at a store.

Main idea

Monomials

Example:4x3 Cubic 5 Constant

Number of terms:One = “mon”4x3

Degree:Sum the exponents of its variables

Main idea

Binomials

Example:7x + 4 Linear9x4 +11 4th

degree

Number of terms:Two = “bi”7x + 4

Degree:The degree of monomial with greatest degree

Main idea

Trinomials

Example:3x2 + 2x + 1 Quadratic

Number of terms:Three =”tri”3x2 + 2x + 1

Degree:The degree of the monomial with greatest degree

Main idea

Polynomials

Example:3x5 + 2x3 + 5x2 + x - 45th degree

Number of terms:Many (> three) =”poly”

Degree: The degree of the monomial with greatest degree


Factor A2 - C2

“DOTS” = Difference of Two Squares

Check for DOTS in DOTS 3(x2 + 4 ) ( x2 - 4)3(x2 + 4 ) ( x + 2)(x - 2)

Check using FOIL.

Step 5

Write the numbers and variables before they were squared in the binomials. (Note: Any even power on a variable is a perfect square. . . just half the exponent when factoring) x2 + 4 ) ( x2 - 4)3(

Step 4

Example: 3x4 - 48Factor out the GCF. 3( x4 – 16)

Check for : 1.) 2 terms 2.) Minus Sign

3.) A and C are perfect squares

Write two binomialsSigns: One + ; One - 3( + ) ( - )

Step 1

Step 2

Step 3


Factoring a quadratic trinomial enables one to determine the x-intercepts of a parabola.

Factoring also enables one to use the x-intercepts to graph.

Function notation /evaluating functions

DomainRange

Function – vertical line test

Relations and

Functions

Point-slope formy – y1 = m(x – x1)

Standard formAx + By = C

Slope-intercept formy = mx + b

Linear Equations

y = yx x

Constant of variation, k

y = kxDirect

Variation

Equation of line of best fit

Scattergram

Line of best fit or trend line

Linear Models

h is horizontal movement and k is vertical movement

Always graphs as a “V”

y = |x – h| + kVertex (h,k)

Absolute Value

Functions

If test is true, shade to include point

Graph line first – called the boundary equation

Test a point not on the line – best choice is (0,0)

Graphing Two-Variable

Inequalities

Functions, Equations, and Graphs

1 Graph the linear equations on the same coordinate plane.

2 If the lines intersect, the solution is the point of intersection.

3 If the lines are parallel, there is no solution.

4 If the lines coincide, there is infinitely many solutions.

1Solve one of the linear equations for one of the variables (look for a coefficient of one).

2Substitute this variable’s value into the other equation.

3Solve the new equation for the one remaining variable.

4Substitute this value into one of the original equations and find the remaining variable value.

1Look for variables with opposite or same coefficients.

2If the coefficients are opposites, add the equations together. If the coefficients are the same, subtract the equations, by changing the sign of each term in the 2nd equation and adding.

3Substitute the value of the remaining variable back into one of the orig. equations to find the other variable.

1 Choose a variable to eliminate.

2 Look the coefficients and find their LCD. This is the value you are trying to get.

3 Multiply each equation by the needed factor to get the LCD.

4 Continue as for regular elimination

GRAPHING SUBSTITUTION ELIMINATION ELINIMATION VIA MULTIPLICATION

The solution (if it has just one) is an ordered pair. This point is a solution to both equations and will test true if substituted into each equation.

Method 1 Method 4Method 3Method 2

SUMMARY

TopicSolving Systems of Linear Equations

Finding solutions for more than one linear equation by using one of four methods.

Main Idea

Details

Five (5)

If a number ends withO or 5

Topic

Details

Divisibility

Dividing numbers that don’t have remainders

Main Idea

Details

Two (2)

If a number ends withO, 2, 4, 6, 8

Main Idea

Details

Ten (10)

If a number ends with O

Main Idea

Details

Nine (9)

If the sum of the digits can be divided by 9

Main Idea

Details

Three (3)

If the sum of the digits can be divided by 3

Exponents are important for the graduation exam. Exponents are also used in exponential functions which model population growth, compound interest, depreciation, radio active decay, and the list goes on an on. . . . .


Exponent Rules

Zero Exponents

n0 = 1-n0 = -1(-n) 0 = 1

Negative Exponents

n-1 = 1/n1/n-1 = n

Multiplying Like Bases

am an = am + n

Power to a Power

(am)n = amn

Product to a Power

(ab)n = an bn

Dividing Like bases

am = am-n

an

Quotient to a Power

(a/b)n = an/ bn

Synonyms Intersection, Overlap, Common, Same

Union, Every, AllMain ideas

Comparing….


Topic Topic

Graph----------│----------------│--------------- -4 0 1

----------│-----│----------│--------------- --2/3 0 5

And Or

To solve real world problems involving chemistry of swimming pool water, temperature, and science.

Notation2x + 3 < 5 and 2x + 3 > -5

-5 < 2x + 3 < 5

3x – 2 > 13 or 3x – 2 < - 4

TOPIC Order of Operations - What we should do in an equation situation…AHSGE: I-1 Apply order of operations

Main Idea

Details

PLEASE

Parenthesis Do all parentheses first.

Main Idea

Details

EXCUSE

ExponentsDo all exponents second.

Main Idea

Details

MY

Multiplication Do all multiplication from left to right next.

Main Idea

Details

DEAR

Division Do all division from left to right next.

Main Idea

Details

AUNT

AdditionDo all addition from left to right.

Main Idea

Details

SALLY

Subtraction Do all subtraction from left to right.

Note: Complete multiplication and division from left to right even if division comes first. Complete addition and subtraction from left to right even if subtraction comes first.

Polynomial ProductsMultiplying Polynomials and Special Products


AHSGE Objective: I -3. Applications include finding area and volume. Special products are used in graphing functions by hand. Punnett Squares in Biology.

Main idea

Polynomial times Polynomial

Monomial times a Polynomial: Use the distributive property Example:-4y2 ( 5y4 – 3y2 + 2 ) = -20y 6 + 12y 4 – 8y2 Polynomial times a Polynomial:Use the distributive PropertyExample: (2x – 3) ( 4x2 + x – 6) = 8x3 – 10x2 – 15x + 18

Main idea

Binomial times Binomial

FOIL: F = FirstO = OuterI = InnerL = Last

Example: (3x – 5 )( 2x + 7)=6x2 + 11 x - 35

Main idea

Square of a Binomial

(a + b) 2 = a2 + 2ab + b2

(a – b) 2 = a2 – 2ab + b2

1st term: Square the first term 2nd term: Multiply two terms and Double 3rd term: Square the last term

Example: (x + 6)2 = x2 + 12 x + 36

Main idea

Difference of Two Squares

DOTS: (a + b) ( a – b) = a2 – b2

Example: (t3 – 6) (t3 + 6) = t6 – 36

These are the steps to …Factor A2 - C2

“DOTS” = Difference of Two Squares


Check for DOTS in DOTS 3(x2 + 4 ) ( x2 - 4)3(x2 + 4 ) ( x + 2)(x - 2)

Check using FOIL.

Step 5

Write the numbers and variables before they were squared in the binomials. (Note: Any even power of a variable is a perfect square. Just half the exponent when factoring) 3(x2 + 4 ) ( x2 - 4)

Step 4

Five Steps

Example: 3x4 - 48Factor out the GCF. 3( x4 – 16)

Check for : 1.) Two terms 2.) Minus Sign 3.) A and C are perfect squares

Write two binomialsSigns: One + ; One - 3( + ) ( - )

Step 1

Step 2

Step 3

Name

AHSGE : I – 4 Factor Polynomials.


Following these steps will allow one to factor any binomial that is not prime.

Factoring a quadratic trinomial enables one to determine the x-intercepts of a parabola.

These are the steps to …Factor Ax2 + Bx + CTrial and Error.


Check using FOIL. 8(3x2 – 3x – x + 1) =8(3x2 – 4x + 1) =24x2 – 32 x + 8

Step 5

If C is positive, determine the factor combination of A and C that will add to give B. If C is negative, determine the factor combination of A and C that will subtract to give B. Since C is positive add to get B: 8 (3x – 1) (x – 1)

Step 4

Five Steps

Example: 24m2 – 32m + 8Factor out the GCF 8(3m2 – 4 m + 1)

Write two binomialsSigns: +C -- signs will be the same sign as the sign of B -C -- signs will be different: one negative and one positive 8( - ) ( - )

List the factors of A and the factors of C.A = 3 C = 1 1, 3 1, 1

Step 1

Step 2

Step 3

Name

AHSGE: I – 4 Factor Polynomials.


Following these steps will allow one to factor any trinomial that is not prime.

Factoring a quadratic trinomial enables one to determine the x-intercepts of the parabola.

These are the steps to …Factoring Polynomials Completely Five Steps

Copyright 2003 Edwin Elliswww.GraphicOrganizers.com

Make sure that each polynomial is factored completely.If you have tried steps 1 – 4 and the polynomial cannot be factored, the polynomial is prime.

Step 5

Four Terms: Grouping Group two terms together that have a GCF. Factor out the GCF from each pair. Look for common binomial. Re-write with common binomial times other factors in a binomial.Example: 5t4 + 20t3 + 6t + 24 = (t + 4) (5t3 + 6)

Step 4

Five Steps

Factor out the GCFExample: 2x3- 6x2= 2x2( x – 3)Always make sure the remaining polynomial(s) are factored.

Two Terms: Check for “DOTS” A2 – C2 (Difference of Two Squares) Example: x2 – 4 = (x – 2 ) (x + 2) See if the binomials will factor again. Check Using FOIL

Three Terms: Ax2 + Bx + C Check for “PST” m2 + 2mn + n2 or m2 – 2mn + n2. Factor using short cut. No “PST”, factor using trial and error. Example PST: c2 + 10 c + 25 = (c + 5)2 For an example of trial and error, see Trial/Error Method.

Step 1

Step 2

Step 3

Name

AHSGE: I – 4Factoring Polynomials.



Factoring allows one to find the x-intercepts and in turn graph the polynomial.

Projectile

Trajectory

Gravity

Vocabulary -- Define and give an example

Acceleration due to gravity

More Vocabulary

Parabola

Fill in the blank. Horizontal and vertical motions are ___________( independent/ dependent) of each other.

When was the snowboard invented?

What is “goofy foot”? What is regular foot?

What is “hang time”?

Snowboarding

Where does hang time occur on a parabola?

Which motion is affected by gravity?

Name sports that involve parabolas.

Snowboarding and its relationship to math

Is about …Big Air Rules

Standard form: ax2 + bx + c = 0

Quadratic Formula

Pythagorean Theorem

c = longest side

Distance Formula

Two points: (x1,y1) (x2,y2)

Two points: (x1,y1) (x2,y2)

MidpointFormula

RADICAL FORMULAS

Most of these formulas involve simplifying a radical.

Topic

Applications: Finding the center of a circle, finding the perimeter of a figure, finding the missing side of a right triangle, solving quadratic equations are all areas where these problems are used in real-world situations.

Q: Why should I know how to use these formulas?A: They are on the exit exam as well as in geometry and higher level math courses.

2 4

2

b b ac

a

2 2 2a b c 2 22 1 2 1( ) ( )x x y y

1 2 1 2,2 2

x x y y

Conditional

An if-then statement If an angle is a straight angle, then its measure is 180.

p → q If p, then q.

Term

Definition Example Symbolic form/read it

Negation (of p)

Has the opposite meaning as the original statement.

An angle is NOT a straight angle.

~pNOT p.

Term


Inverse

Negates both the if and then of a conditional statement

If an angle is NOT a straight angle, then its measure is NOT 180.

~p → ~qIf NOT p, then NOT q.

Term


Contra-positive

Switches the if and then and negates both.

If an angle’s measure is NOT 180, then it is NOT a straight angle.

~q → ~pIf NOT q, then NOT p.

Term


Name ___________________Conditional

An if-then statement If an angle is a straight angle, then its measure is 180.

The part following the if is the hypothesis and the part following the then is the conclusion.

p → qIf p, then q.

Term


Converse

Switches the if and then of a conditional statement

If the measure of an angle is 180, then it is a straight angle.

q → pIf q, then p.

Term


Bi-conditional

The combination of a conditional statement and its converse; usually contains the words if and only if.

An angle is a straight angle if and only if its measure is 180.

p ↔ qp if and only if q.

Term


Term

Definition Picture Knowledge Connection

The ratio of the sides is 1 : 2: √3

Side opposite the 30º angle = ½ hypotenuse

Side opposite the 60º angle = ½ hypotenuse times √3

The larger leg equals the shorter leg times √3

30º- 60º- 90º

45º- 45º- 90º

The ratio of the sides is 1 : 1 : √2

Side opposite the 45º angle = ½ hypotenuse times √2

Hypotenuse = s √2 where s = a leg

Is about …Special Right Triangles

30°

60°

45°

45°

Math Curseby Jon Scieszka + Lane Smith

How do you get to school in the morning? What time do you leave and what time do you arrive? If you were 15 minutes late leaving your house for school, what time would you arrive? What would you not have time to do if you were late and why?

If there were approximately 1300 students in the school, how many fingers would there be? How would you write that in scientific notation?

If an M&M is about a centimeter long, then how many M&Ms long is your foot?

If he bought an 80cent candy bar that was on sale for 25% off, how much would he have to pay? Would it be more or less than the candy bar that was on for sale for 50% off? Explain.

Title

1.

2.

3.

4.

Writing Linear Equations …

Three Forms of a Linear Equation

Main Idea

Details

1. Given slope = m , and y-int= b

2. Substitute m and b into the

equation.

3. Transform to Standard if necessary.

Slope-Intercept Form

Y = mx + b

Main Idea

Details

1. Given an equation in slope-

intercept form. a. Eliminate fractions b. Add or Subtract

2. Given an equation in point-slope form.

a. Distribute b. Eliminate Fractions c. Add or Subtract

Standard Form

Ax + By = C

Main Idea

Details

1. Given slope = m&point (x1, y1)

Substitute m and the point into the equation.

2. Given 2 points a. Find m b. Substitute into point slope form c. Simplify/change to

Slope- intercept or Standard form if necessary.

Point-Slope Form

y – y1 = m(x – x1)


Many real life situations can be described by an equation, for instance, payroll deductions, temperature . . .

Documents

GeometricGeometric ADVANCED ALG/TRIG Chapter 11 – Sequences and Series Sequences (an ordered list of numbers) Geometric R = common ratio Geometric mean