# Sequences, Probability, and Counting Theory

## Arithmetic Sequences


### Learning Objectives

1. Determine if a sequence is arithmetic (IA 12.2.1) 
2. Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)

### Objective 1: Determine if a sequence is arithmetic (IA 12.2.1)

An arithmetic sequence is a sequence where the difference between consecutive terms is always the same.

The difference between consecutive terms, d, and is called the common difference, for n greater than or equal to two.

    

### Practice Makes Perfect 

Determine if each sequence is arithmetic. If so, indicate the common difference.


### Practice Makes Perfect


### Objective 2: Find the general term (nth term) of an arithmetic sequence (IA 12.2.2)

In the last section, we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is  and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with  .
The first term adds 0d to the  , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to  is one less than the number of the term. We then have the formula for the general term of an arithmetic sequence.

### Practice Makes Perfect


Find the general term (nth term) of an arithmetic sequence.

Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.
As an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider specific kinds of sequences that will allow us to calculate depreciation, such as the truck’s value.

### Finding Common Differences

The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.
The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term.

### Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of 
 and 
 into formula below.

### Using Recursive Formulas for Arithmetic Sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

### Using Explicit Formulas for Arithmetic Sequences

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.



The common difference is 
 , so the sequence represents a linear function with a slope of 
 . To find the 
 -intercept, we subtract 
 from 
  . You can also find the 
 -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in .
Recall the slope-intercept form of a line is 
  When dealing with sequences, we use 
  in place of 
  and 
  in place of 
  If we know the slope and vertical intercept of the function, we can substitute them for 
  and 
  in the slope-intercept form of a line. Substituting 
  for the slope and 
  for the vertical intercept, we get the following equation:
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is 
 , which simplifies to 
 


### Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

### Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term of 
 instead of 
 In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

### Key Equations


### Key Concepts

1. An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
2. The constant between two consecutive terms is called the common difference.
3. The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See .
4. The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See  and .
5. A recursive formula for an arithmetic sequence with common difference 
  is given by 
  See .
6. As with any recursive formula, the initial term of the sequence must be given.
7. An explicit formula for an arithmetic sequence with common difference 
  is given by 
  See .
8. An explicit formula can be used to find the number of terms in a sequence. See .
9. In application problems, we sometimes alter the explicit formula slightly to 
  See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, find the common difference for the arithmetic sequence provided.
For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.
For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.
For the following exercises, write the first five terms of the arithmetic series given two terms.
For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.
For the following exercises, find the first term given two terms from an arithmetic sequence.
For the following exercises, find the specified term given two terms from an arithmetic sequence.
For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.
For the following exercises, write a recursive formula for each arithmetic sequence.
For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.
For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.
For the following exercises, write an explicit formula for each arithmetic sequence.
For the following exercises, find the number of terms in the given finite arithmetic sequence.

### Graphical

For the following exercises, determine whether the graph shown represents an arithmetic sequence.
For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.

### Technology

For the following exercises, follow the steps to work with the arithmetic sequence 
  using a graphing calculator:


For the following exercises, follow the steps given above to work with the arithmetic sequence 
  using a graphing calculator.

### Extensions
