Infinite Geometric Series
Definition of an Infinite Geometric Series
We learned that a geometric series has the form
is called the infinite geometric series.
Calculating the Infinite Geometric Series
Suppose that a run
Sequences and Series
Example: Find the next term and describe the pattern:
2, 4, 6, 8, 10, .
1, 4, 9, 16, 25, .
3, 7, 15, 31, 63, .
1, -1/2, 1/6, -1/24, 1/120, -1/720, .
We see that the next term is 12. We can get to
A conic section is formed by intersecting a plane with a cone. The different possible
conic sections are the circle, parabola, ellipse, and the hyperbola.
A circle is the set of points in a plane a fixed distance f
Definition of the Ellipse (Geometric)
Let P and Q be two points (the foci) in the plane. The ellipse is the set of all points R in
the plane such that PR + QR is a fixed constant. An ellipse can be constructed using a
piece of string. Fix t
The Definition of the Logarithm
The function logbx is defined as the inverse function of y = bx
Recall that by definition, if f and g are inverse functions then
f(g(x) = g(f(x) = x
Hence we have the following two properties:
Properties of Logarithms
Properties of Logarithms and their proofs
logbxy = ylogbx
logbxy = logb(blogb(x)y
= logb(bylogb(x) = ylogbx
logb(xy) = logbx + logby
logb(x/y) = logbx - logby
Linear Programming (An Example)
P = 2x + 5
subject to the constraints
x + 3y < 15
4x + y < 16
First we graph the system of inequalities.
x + 3y = 15
we use (0,5) and (15,0) and note that the arrows point
Geometry of Systems of Equations
We know that for two by two linear systems of equation, the geometry is that of two lines
that either intersect, are parallel, or are the same line. If they intersect then there is
exactly one solution, i
Definition of a Matrix
An m by n matrix is an array of numbers with m rows and n columns.
is a 3 by 2 matrix.
Consider the system of equations
2x - y + 3z = 5
4z = 3
5x - 7y + 3z = 7
Then the matrix
Definition of the Hyperbola (Geometric)
The final conic that we will study is called the hyperbola.
Let P and Q be two fixed points, and c be a constant. Then the
set of points in the plane such that
|QR - PR| = c
is a hyperbol
Algebraic Definition of The Parabola
Recall that the standard equation of the parabola is given by
y = a(x - h)2 + k
If we are given the equation of a parabola
y = ax2 +bx + c
we can complete the square to get the parabola in standard form.
Factorials and Their Applications
Definition of the Factorial
We define n! recursively by
0! = 0,
1! = 1,
n! = n(n - 1)!
5! = 5(4)(3)(2) = 120
Suppose that we are interested in how many ways there are in scrambling the letters of
Geometric Sequences And Series
Example: Find the General Element
A) 3, 6, 12, 24, 48, .
B) 5, 15, 45, 135, .
C) -3, 30, -300, 3000, .
D) 2, 2/3, 2/9, 2/27, .
We see that to get to the next t
Evaluating a Series
If we have a finite series there are two ways of evaluating it. The first way is
computation by hand.
n = 15 (2n-1)
1 + 3 + 5 + 7 + 9 = 25.
Notice that we just plugged in the values 1,2,3,4, and 5 for n and add
Example of an Exponential Function
A biologist grows bacteria in a culture. If initially there were three grams of bacteria,
after one day there are six grams of bacteria, and after two days, there are twelve grams,
how many grams will the
Exponential and Log Equations
Equations that Involve Logs
Step by Step Method
Step 1: Contract to a single log.
Step 2: Get the log by itself.
Step 3: Exponentiate both sides with the appropriate base.
Step 4: Solve.
Step 5: Check your solution for dom
Arithmetic Sequences and Series
Find the general term for the following sequences both recursively and explicitly:
Determinants and Inverses
Consider row reducing the standard 2x2 matrix. Suppose that a is nonzero.
1/a R1 -> R1
R2 - cR1 -> R2
0 d - cb/a
Now notice that we cannot make the lower right corner a 1 if
d - cb/a = 0