Logarithms
If a > 1 or 0 < a < 1, then the exponential function f : R (0, ) dened
as f (x) = ax is one-to-one and onto. That means it has an inverse function.
If either a > 1 or 0 < a < 1, then the inverse of the function ax is
loga : (0, ) R
and its call
Rules for Numbers
Rules for Numbers
The real numbersRules by a a collection of rules that have to do with
are governed
The real numbers are governed forcollection of rules that have to do with
Rules by Numbers
addition, multiplication, and inequalities. I
Sequences
A sequence is an innite list of numbers.
Sequences are written in the form
a1 , a2 , a3 , a4 , .
where a1 R, and a2 R, and a3 R, and a4 R, and so on.
A shorter way to write whats above is to say that a sequence is an innite
list
a1 , a2 , a3 , a
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Practice Final Exam
Discrete math
4
(i2 1)
1.) Find
i=1
2.) Find
i=1
2
6i
3.) What is the 61st term of the sequence 7, 11, 15, 19, .?
4.) Whats the 57th term of 3, 6, 12, 24, .?
5.) Whats the sum of the rst 60 terms of the sequence 3, 5, 7, 9, . . .?
31
Counting II
Sometimes we will want to choose k objects from a set of n objects, and
we wont be interested in ordering them. For example, if you are leaving for
vacation and you want to pack your suitcase with three of the seven pairs
of shorts that you ow
Sums & Series
Suppose a1 , a2 , . is a sequence.
Sometimes well want to sum the rst k numbers (also known as terms)
that appear in a sequence. A shorter way to write a1 + a2 + a3 + + ak is as
k
ai
i=1
There are four rules that are important to know when u