# Sequences, Probability, and Counting Theory

## Sequences and Their Notations


### Learning Objectives

1. Write the first few terms of a sequence (IA 12.1.1)
2. Find a formula for the general term (nth term) of a sequence (IA 12.1.2)

### Objective 1: Write the first few terms of a sequence (IA 12.1.1).

A patient takes a 30 mg antibiotic capsule. At the end of that hour, the amount of antibiotic remaining in her body is only 90% of the amount in the beginning of that hour. The 30mg dose is taken at time t = 1 hour. How much of this dose remains at the end of 1 hour? 2hours? 3 hours? 4 hours?
This ordered list of numbers 27, 24.3, 21.87, 19.68, … is a sequence. Each number in the list is a term.

A sequence is a function whose domain is the counting numbers. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an infinite sequence.
Often when working with sequences we do not want to write out all the terms. We want a more compact way to show how each term is defined. When we worked with functions, we wrote  and we said the expression 2x was the rule that defined values in the range.

While a sequence is a function, we do not use the usual function notation. Instead of writing the function as  , we would write it as  . The  is the , the term in the nth position where n is a value in the domain. The formula for writing the nth term of the sequence is called the general term or formula of the sequence.

General sequence terms are denoted as follows:  

### Practice Makes Perfect


Write the first few terms of a sequence.


### Objective 2: Find a formula for the general term (nth term) of a sequence (IA 12.1.2)

Sometimes we have a few terms of a sequence and it would be helpful to know the general term or . To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.


### Practice Makes Perfect

A video game company launches an exciting new advertising campaign. They predict the number of online visits to their website, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the third day, and so on. See .
If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of ordered lists.

### Writing the Terms of a Sequence Defined by an Explicit Formula

One way to describe an ordered list of numbers is as a sequence. A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is
The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term. The first five terms of this sequence are 2, 4, 8, 16, and 32.
Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.
One type of formula is an explicit formula, which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We can use the formula to find the nth term of the sequence, where 
  is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the 
  term.

The first term of the sequence is 
  the second term is 
  the third term is 
  and so on. The 
  term of the sequence can be found by raising 2 to the 
  power. An explicit formula for a sequence is named by a lower case letter 
  with the subscript 
  The explicit formula for this sequence is
Now that we have a formula for the 
  term of the sequence, we can answer the question posed at the beginning of this section. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To find the number of hits on the last day of the month, we need to find the 31st term of the sequence. We will substitute 31 for 
  in the formula.
If the doubling trend continues, the company will get 
  hits on the last day of the month. That is over 2.1 billion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions.
Another way to represent the sequence is by using a table. The first five terms of the sequence and the 
  term of the sequence are shown in .
Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in  that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function.
Lastly, we can write this particular sequence as
A sequence that continues indefinitely is called an infinite sequence. The domain of an infinite sequence is the set of counting numbers. If we consider only the first 10 terms of the sequence, we could write
This sequence is called a finite sequence because it does not continue indefinitely.

### Investigating Alternating Sequences

Sometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to finding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show increase or decrease as 
  increases. Let’s take a look at the following sequence.
Notice the first term is greater than the second term, the second term is less than the third term, and the third term is greater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret the sequence.

### Investigating Piecewise Explicit Formulas

We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions. Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.

### Finding an Explicit Formula

Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the 
  term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.

### Writing the Terms of a Sequence Defined by a Recursive Formula

Sequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern of the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term can be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,…. Other examples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.
Each term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be written using an explicit formula. Instead, we describe the sequence using a recursive formula, a formula that defines the terms of a sequence using previous terms.
A recursive formula always has two parts: the value of an initial term (or terms), and an equation defining 
  in terms of preceding terms. For example, suppose we know the following: 
We can find the subsequent terms of the sequence using the first term.
So the first four terms of the sequence are 
  .
The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms. 
To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so 

### Using Factorial Notation

The formulas for some sequences include products of consecutive positive integers.  , written as 
  is the product of the positive integers from 1 to 
  For example, 
An example of formula containing a factorial is 
  The sixth term of the sequence can be found by substituting 6 for 
 

The factorial of any whole number 
  is 
  We can therefore also think of 
  as 
  

### Key Equations


### Key Concepts

1. A sequence is a list of numbers, called terms, written in a specific order.
2. Explicit formulas define each term of a sequence using the position of the term. See , , and .
3. An explicit formula for the 
  term of a sequence can be written by analyzing the pattern of several terms. See .
4. Recursive formulas define each term of a sequence using previous terms.
5. Recursive formulas must state the initial term, or terms, of a sequence.
6. A set of terms can be written by using a recursive formula. See  and .
7. A factorial is a mathematical operation that can be defined recursively.
8. The factorial of 
  is the product of all integers from 1 to 
  See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, write the first four terms of the sequence.
For the following exercises, write the first eight terms of the piecewise sequence.
For the following exercises, write an explicit formula for each sequence.
For the following exercises, write the first five terms of the sequence.
For the following exercises, write the first eight terms of the sequence.
For the following exercises, write a recursive formula for each sequence.
For the following exercises, evaluate the factorial.
For the following exercises, write the first four terms of the sequence.

### Graphical

For the following exercises, graph the first five terms of the indicated sequence
For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.
For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

### Technology

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

For the following exercises, use the steps above to find the indicated term or terms for the sequence.
Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

Using a TI-83, do the following.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

### Extensions
