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Math 2200 Chapter 1 Arithmetic and Geometric Sequences and ... · PDF file1 Math 2200 Chapter 1 Arithmetic and Geometric Sequences and Series Review Key Ideas Description or Exampl

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    Math 2200 Chapter 1 Arithmetic and Geometric Sequences and Series Review

    Key Ideas Description or Example

    Sequences An ordered list of numbers where a mathematical pattern can be used to determine the next terms.

    Example: 1, 5, 9, 13, 17... or 1000, 100, 10, 1... n is the term position or the number of terms, n must be a natural number

    Series The sum of all the terms of a finite sequence Example: 5 + 10 + 15 + 20

    1 + 0.5 + 0.25 + 0.125...


    Sequence A sequence that has a common difference, 1n nd t t

    Example: 2, 4, 6, 8, 10, 12, 14,... where d = 2

    Graph of an



    Always discrete since the n values or the term position must be natural numbers.

    Related to a linear function y = mx + b where m = d and 1b t m .

    The slope of the graph represents the common difference of the general term of the sequence.

    t1 = b + m , add the y-intercept to the slope to get the value of the first term of the sequence.


    Series The sum of an arithmetic sequence.

    Use 12

    n n

    nS t t when you know the first term, last term, and the number of terms.

    Use 12 12


    nS t n d when you know the first term, the common difference,

    and the number of terms.

    You may need to determine the number of terms by using 1 1nt t n d .


    Sequence A sequence that has a common ratio. 11


    nt t r

    Example: 3, 9, 27, 82, 243, 729, 2187... where r = 3

    Graph is discrete, not continuous, and not linear.



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    Finite Geometric


    A finite geometric series is the expression for the sum of the terms of a finite geometric


    The General formula for the Sum of the first n terms

    Known Values are: Known Values are:

    t1, r and n t1, r and tn


    Infinite Geometric


    A geometric series that does not end or have a final term.

    It may be convergent (sum approaches a value, there is a formula for this) or divergent

    (sum gets infinitely larger).

    The series is convergent if the absolute value of r is a fraction or decimal less than one:

    |r|1, -1 > r > 1

    Sum of an Infinite

    Geometric Series,

    only if it


    You must know the value of the first term and the common ratio.

    General or explicit

    formula The unique parameters are substituted into the formula. The parameters are 1t and r for a geometric sequence or d for arithmetic sequence.

    Example: 13 2nnt or 2 3nt n

    1( 1) , 11



    t rS r


    1 , 11


    rt tS r


    1 , 11

    tS r


  • 3

    Math 2200 Chapter 2 Trigonometry Review

    Key Ideas Description or Example

    Sketching an angle in standard position.

    The measure of an angle in standard position can be

    between 0 to 360 .

    The symbol is often used to represent the measure

    of an angle.

    The vertex of the angle is located at the origin (0,0) on a

    Cartesian plane.

    The initial arm of the angle lies along the positive x-axis.

    The terminal arm rotates in a positive direction.

    The angle in standard position is measured from the

    positive x-axis.

    Each angle in standard position has a corresponding

    reference angle.

    Reference angles are positive acute angles (< 90)

    measured from the terminal arm to the nearest x-


    Any angle from 90 to 360 is the reflection in the

    x- axis and/or the y-axis of its reference angle.

    Consider a reference right angle triangle

    The three primary trigonometry ratios are

    To calculate a ratio given an angle using your


    The mode of your calculator should be in degrees.

    The ratios are given as decimal approximations.

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    Given any point P(x, y) on the Terminal Arm of an

    angle in standard position, the Pythagorean

    Theorem can be used to determine the distance the

    point is from the origin.

    This distance can be labeled r.

    The x- and y- coordinates of the point can be used to

    determine the exact values for the primary trig


    The points P(x, y), P(x, y), P(x, y) and P(x, y)

    are points on the terminal sides of angles in standard

    position that have the same reference angle. These

    points are reflections of the point P( x, y) in the x-

    axis, y-axis or both axes.

    The trigonometry ratios may be positive or negative

    in value depending on which quadrant the terminal

    arm is in. A point on the terminal arm would have

    coordinates (x, y).

    A Smart Trig Class

    There are two special right triangles for which you

    can determine the exact values of the primary

    trigonometric ratios.

    TIP: the smallest angle is always opposite the

    shortest side.

    Determining the measure of an angle, given the

    defining ratio.

    Use the inverse trig ratios.

    Quadrantal Angles

    If the terminal arm lies on an axis, the angle is

    called a Quadrantal angle (it separates the


    The quadrants are labeled in a counterclockwise


    The Quadrantal angles for one revolution are

    0, 90, 180, 270, 360

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    Ratios of Quadrantal Angles

    To determine, without the use of technology, the

    value of sin , cos or tan , given any point

    P (x, y) on the terminal arm of angle , where = 0,

    90, 180, 270 or 360 use the defining ratios and

    consider any point on the axis.

    Plot the point, determine the values of x, y, and r and

    calculate the ratio.

    The Sine Law

    An angle and its opposite side create a defining ratio

    that can be used to calculate the other


    Notice you are given the measurements of an angle

    and its opposite side.

    You may be asked to calculate the measure of the

    remaining sides or angles.

    sin sin sin

    a b c

    A B C

    The Ambiguous Case of the Sine Law

    Used when you are finding an Angle.

    Given: two angles and one side or

    two sides and an angle opposite one of the given


    An angle and its opposite side are given to define

    one ratio sin


    A of the Sine Law.

    Since angle B could be obtuse or acute, the

    ambiguous case must be considered.

    There are two possible measure for angle B.

    The Cosine Law

    Describes the relationship between the cosine of an

    angle and the lengths of the three sides of any


    2 2 22 2 2 2 cos or cos


    b c aa b c bc A A


    To determine side c


    sin31 sin104


    20(sin104 )


    c= 37.7

  • 6

    Math 2200 Chapter 3 Quadratic Functions Review

    Key Ideas Description or Example

    Quadratic Functions

    polynomial of degree two Standard Form: f(x) = ax

    2 + bx + c, a 0

    Vertex Form: f(x) = a(x p)2 + q , a 0

    Factored Form: f(x) = (x + c)( x + d)

    For a quadratic function, the graph is

    in the shape of a parabola

    Parent Graph is y = x2

    Characteristics of a Quadratic

    Function from the vertex form of the


    f(x) = a(x - p)2 + q

    The coordinates of the vertex are at (p, q). Note that the negative symbol from (x - p) does not transfer to the value of p.

    f(x) = 2(x - 3)2 + 4 vertex at (3, 4)

    f(x) = 2(x + 3)2 + 4 vertex at (-3, 4)

    Horizontal Translations:

    When p > 0 the graph moves (translated) to the right.

    When p < 0 is translated or shifts to the left.

    Vertical Translations:

    When q > 0 the parabola shifts up.

    When q < 0 the parabola shifts down.

    The parameter a indicates the direction of opening as well as how narrow or wide the graph is in relation to the parent graph.

    When a > 0 , the graph opens up and the vertex is a minimum

    min max

    When a < 0, the graph opens downward and the vertex is a maximum.

    When -1 < a < 1, the parabola is wider than the parent graph.

    When a > 1 or a < -1, the parabola is narrower than the parent graph.

    The Axis of Symmetry is an imaginary vertical line through the vertex that divides the function graph into two symmetrical parts. The

    equation for the axis of symmetry is represented by x p = 0 or x = p

    The domain of a quadratic function is all real numbers |x x . The only exception is for real life models.

    The range of a quadratic function depends on the value of the parameters a and q.

    When a is positive, the range is |y y q .

    When a is negative, the range is |y y q .

    x- and y- intercepts Calculate the x-intercept pt (x, 0) by replacing y with 0 in either form of

    the function equation. ax2 + bx + c = 0 or a(x p)

    2 + q = 0.

    Calculate the y-intercept (0, y) by replacing x with 0 and solving for the

    value of y.

    Summary of Characteristics.

    Write a quadratic function in the form

    y = a(x p)2 + q for a given graph or

    a set of characteristics of a graph.