# 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


# 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


# Sequences, Probability and Counting Theory

## Geometric Sequences

### Learning Objectives

1. Determine if a sequence is geometric (IA 12.3.1).
2. Find the general term (nth term) of a geometric sequence (IA 12.3.2).

### Objective 1: Determine if a sequence is geometric (IA 12.3.1)

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same.

The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.

### Practice Makes Perfect

Determine if each sequence is geometric. If so, indicate the common ratio.

### Practice Makes Perfect

### Objective 2: Find the general term (nth term) of a geometric sequence (IA 12.3.2)

Let’s find the formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is  and the common ratio is  . We will then look for a pattern.

### Practice Makes Perfect

Find the general term (nth term) of a geometric sequence.

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.

### Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

### Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is
  and the common ratio is
  we can find subsequent terms by multiplying
  to get
  then multiplying the result
  to get
  and so on.
The first four terms are

### Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

### Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.
Let’s take a look at the sequence
  This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is
The graph of the sequence is shown in .

### Solving Application Problems with Geometric Sequences

In real-world scenarios involving geometric sequences, we may need to use an initial term of
  instead of
  In these problems, we can alter the explicit formula slightly by using the following formula:

### Key Equations

### Key Concepts

1. A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
2. The constant ratio between two consecutive terms is called the common ratio.
3. The common ratio can be found by dividing any term in the sequence by the previous term. See .
4. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See  and .
5. A recursive formula for a geometric sequence with common ratio
  is given by
  for
  .
6. As with any recursive formula, the initial term of the sequence must be given. See .
7. An explicit formula for a geometric sequence with common ratio
  is given by
  See .
8. In application problems, we sometimes alter the explicit formula slightly to
  See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the common ratio for the geometric sequence.
For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.
For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.
For the following exercises, write the first five terms of the geometric sequence, given any two terms.
For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.
For the following exercises, find the specified term for the geometric sequence, given the first four terms.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write a recursive formula for each geometric sequence.
For the following exercises, write the first five terms of the geometric sequence.
For the following exercises, write an explicit formula for each geometric sequence.
For the following exercises, find the specified term for the geometric sequence given.
For the following exercises, find the number of terms in the given finite geometric sequence.

### Graphical

For the following exercises, determine whether the graph shown represents a geometric sequence.
For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

### Extensions


# Sequences, Probability and Counting Theory

## Series and Their Notations

### Learning Objectives

1. Use summation notation to write a sum. (IA 12.1.5)
2. Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Objective 1: Use summation notation to write a sum. (IA 12.1.5)

A series is the sum of the terms of a sequence. For example,  1 + 6 + 11+ 16 + 21 + 26 + 31 is the sum of the first seven terms arithmetic sequence with general term,
We write a series by using the summation notation. In order to write that summation, we will need to find the general term of our sequence and the summation will look like:

For the series, 1 + 6 + 11 + 16  + 21 + 26 + 31 + .... the summation notation is

### Practice Makes Perfect

Use summation notation to write the sum.

### Objective 2: Find the sum of the first n terms of an arithmetic sequence. (IA 12.2.3)

### Practice Makes Perfect

A parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the amount of money invested and the amount of interest earned.

### Using Summation Notation

To find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the amounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called a series. Consider, for example, the following series.
The  of a series is the sum of a finite number of consecutive terms beginning with the first term. The notation
 represents the partial sum.
Summation notation is used to represent series. Summation notation is often known as sigma notation because it uses the Greek capital letter sigma,
  to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, called the upper limit of summation, is the number used to generate the last term in a series.
If we interpret the given notation, we see that it asks us to find the sum of the terms in the series
  for
  through
  We can begin by substituting the terms for
 and listing out the terms of this series.
We can find the sum of the series by adding the terms:

### Using the Formula for Arithmetic Series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference,
 The sum of the terms of an arithmetic sequence is called an arithmetic series. We can write the sum of the first
 terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first
  terms of an arithmetic series, we can derive a formula for the sum of the first
  terms of any arithmetic series.
Because there are
 terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first
  terms of an arithmetic series.
Use the formula to find the sum of each arithmetic series.

### Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio,
  We can write the sum of the first
  terms of a geometric series as
Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first
  terms of a geometric series. We will begin by multiplying both sides of the equation by

Next, we subtract this equation from the original equation.
Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for
 divide both sides by

Use the formula to find the indicated partial sum of each geometric series.

### Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first
 terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is

This series can also be written in summation notation as
 where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

### Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric sequence approach 0, the sum of an infinite geometric series can be defined.  The terms in this series approach 0:
The common ratio

 As
  gets very large, the values of
 get very small and approach 0.   Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with
 approach 0; the sum of a geometric series is defined when

Determine whether the sum of the infinite series is defined.

### Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first
 terms of a geometric series.
We will examine an infinite series with
 What happens to
  as
  increases?
The value of
  decreases rapidly. What happens for greater values of

As
  gets very large,
 gets very small. We say that, as
  increases without bound,
 approaches 0. As
  approaches 0,
 approaches 1. When this happens, the numerator approaches
  This give us a formula for the sum of an infinite geometric series.
Find the sum, if it exists.

### Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.
We can find the value of the annuity right after the last deposit by using a geometric series with
 and
 After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.
We can find the value of the annuity after
 deposits using the formula for the sum of the first
 terms of a geometric series. In 6 years, there are 72 months, so
 We can substitute
 into the formula, and simplify to find the value of the annuity after 6 years.
After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of $50 each for a total of
  This means that because of the annuity, the parent earned $720.44 interest in their college fund.

### Key Equations

### Key Concepts

1. The sum of the terms in a sequence is called a series.
2. A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See .
3. The sum of the terms in an arithmetic sequence is called an arithmetic series.
4. The sum of the first
 terms of an arithmetic series can be found using a formula. See  and .
5. The sum of the terms in a geometric sequence is called a geometric series.
6. The sum of the first
 terms of a geometric series can be found using a formula. See  and .
7. The sum of an infinite series exists if the series is geometric with

8. If the sum of an infinite series exists, it can be found using a formula. See , , and .
9. An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, express each description of a sum using summation notation.
For the following exercises, express each arithmetic sum using summation notation.
For the following exercises, use the formula for the sum of the first
 terms of each arithmetic sequence.
For the following exercises, express each geometric sum using summation notation.
For the following exercises, use the formula for the sum of the first
  terms of each geometric sequence, and then state the indicated sum.
For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason.

### Graphical

For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.
For the following exercises, use the geometric series

### Numeric

For the following exercises, find the indicated sum.
For the following exercises, use the formula for the sum of the first  terms of an arithmetic series to find the sum.
For the following exercises, use the formula for the sum of the first
  terms of a geometric series to find the partial sum.
For the following exercises, find the sum of the infinite geometric series.
For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate.

### Extensions

### Real-World Applications


# Sequences, Probability and Counting Theory

## Counting Principles

### Learning Objectives

1. Solve counting problems using the addition principle.
2. Solve counting problems using the multiplication principle.

### Objective 1: Solve counting problems using the addition principle.

In probability theory, an outcome is a possible result of an experiment or trial.

In probability theory, an event is a set of outcomes of an experiment.

Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.

The addition principle applies when we are making only one selection.

### Practice Makes Perfect

Solve counting problems using the addition principle.

### Objective 2: Solve counting problems using the multiplication principle.

The Multiplication Principle applies when we are making more than one selection.

### Practice Makes Perfect

Solve counting problems using the multiplication principle.

A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a monogram. The customer can choose the order of the images and the letters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the huge number of options they offer. Counting the possibilities is challenging!
We encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of counting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes, and college scheduling choices. We will examine this type of mathematics in this section.

### Using the Addition Principle

The company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet models and 5 supported smartphone models. The Addition Principle tells us that we can add the number of tablet options to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8 total options, as we can see in .

### Using the Multiplication Principle

The Multiplication Principle applies when we are making more than one selection. Suppose we are choosing an appetizer, an entrée, and a dessert. If there are 2 appetizer options, 3 entrée options, and 2 dessert options on a fixed-price dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram in .
The possible choices are:
1. soup, chicken, cake
2. soup, chicken, pudding
3. soup, fish, cake
4. soup, fish, pudding
5. soup, steak, cake
6. soup, steak, pudding
7. salad, chicken, cake
8. salad, chicken, pudding
9. salad, fish, cake
10. salad, fish, pudding
11. salad, steak, cake
12. salad, steak, pudding
We can also find the total number of possible dinners by multiplying.
We could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.

### Finding the Number of Permutations of n Distinct Objects

The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.

### Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle

To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.

There are four options for the first place, so we write a 4 on the first line.

After the first place has been filled, there are three options for the second place so we write a 3 on the second line.

After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.

There are 24 possible permutations of the paintings.
A family of five is having portraits taken. Use the Multiplication Principle to find the following.

### Finding the Number of Permutations of n Distinct Objects Using a Formula

For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of
  objects and we want to choose
  objects from the set in order, we write
  Another way to write this is
  a notation commonly seen on computers and calculators. To calculate
  we begin by finding
  the number of ways to line up all
  objects. We then divide by
  to cancel out the
  items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is
  Using factorials, we get the same result.
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
Note that the formula stills works if we are choosing all
  objects and placing them in order. In that case we would be dividing by
  or
  which we said earlier is equal to 1. So the number of permutations of
  objects taken
  at a time is
  or just

A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.

### Find the Number of Combinations Using the Formula

So far, we have looked at problems asking us to put objects in order. There are many problems in which we want to select a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the order does not matter, we are dealing with combinations. A selection of
  objects from a set of
  objects where the order does not matter can be written as
  Just as with permutations,
  can also be written as
  In this case, the general formula is as follows.
An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select 3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways to order 3 paintings. There are
  ways to order 3 paintings. There are
  or 4 ways to select 3 of the 4 paintings. This number makes sense because every time we are selecting 3 paintings, we are not selecting 1 painting. There are 4 paintings we could choose not to select, so there are 4 ways to select 3 of the 4 paintings.

### Finding the Number of Subsets of a Set

We have looked only at combination problems in which we chose exactly
  objects. In some problems, we want to consider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings. Any number of toppings can be ordered. How many different pizzas are possible?
To answer this question, we need to consider pizzas with any number of toppings. There is
  way to order a pizza with no toppings. There are
  ways to order a pizza with exactly one topping. If we continue this process, we get
There are 32 possible pizzas. This result is equal to

We are presented with a sequence of choices. For each of the
  objects we have two choices: include it in the subset or not. So for the whole subset we have made
  choices, each with two options. So there are a total of
  possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time.

### Finding the Number of Permutations of n Non-Distinct Objects

We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be  ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the objects are not distinct, many of the
  permutations we counted are duplicates. The general formula for this situation is as follows.
In this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find the number of unique permutations of the stickers. There are
  ways to order the stars and
  ways to order the moon.
There are 3,326,400 ways to order the sheet of stickers.

### Key Equations

### Key Concepts

1. If one event can occur in
  ways and a second event with no common outcomes can occur in
  ways, then the first or second event can occur in
  ways. See .
2. If one event can occur in
  ways and a second event can occur in
  ways after the first event has occurred, then the two events can occur in
  ways. See .
3. A permutation is an ordering of
  objects.
4. If we have a set of
  objects and we want to choose
  objects from the set in order, we write

5. Permutation problems can be solved using the Multiplication Principle or the formula for
  See  and .
6. A selection of objects where the order does not matter is a combination.
7. Given
  distinct objects, the number of ways to select
  objects from the set is
  and can be found using a formula. See .
8. A set containing
  distinct objects has
  subsets. See .
9. For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See .

### Section Exercises

### Verbal

For the following exercises, assume that there are
  ways an event
  can happen,
  ways an event
  can happen, and that
  are non-overlapping.
Answer the following questions.

### Numeric

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations.
For the following exercises, compute the value of the expression.
For the following exercises, find the number of subsets in each given set.
For the following exercises, find the distinct number of arrangements.

### Extensions

### Real-World Applications


# Sequences, Probability and Counting Theory

## Binomial Theorem

### Learning Objectives

1. Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

### Objective 1: Use Pascal’s Triangle to expand a binomial. (IA 12.4.1)

Pascal’s triangle helps us find the coefficients of the terms in the expansion of a binomial.

To find the coefficients of the terms, we write our expansion again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients. The array to the right is called Pascal’s Triangle.

Notice that in each expansion the powers of a in each term decrease from n to 0, and the powers of b increase from 0 to n.

Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.

To find the coefficients of the expansion of the binomial  , go to the row that has the value n as a second entry.

### Practice Makes Perfect

Use Pascal’s Triangle to expand a binomial.

A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find
  without multiplying the binomial by itself
  times.

### Identifying Binomial Coefficients

In Counting Principles, we studied combinations. In the shortcut to finding
  we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation
  instead of
  but it can be calculated in the same way. So
The combination
  is called a binomial coefficient. An example of a binomial coefficient is

### Using the Binomial Theorem

When we expand
  by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand
  we might multiply
  by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.
First, let’s examine the exponents. With each successive term, the exponent for
  decreases and the exponent for
  increases. The sum of the two exponents is
  for each term.
Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:
These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.
Another way to see the coefficients is to examine the expansion of a binomial in general form,
  to successive powers 1, 2, 3, and 4.
Can you guess the next expansion for the binomial

See , which illustrates the following:
1. There are
  terms in the expansion of

2. The degree (or sum of the exponents) for each term is

3. The powers on
  begin with
  and decrease to 0.
4. The powers on
  begin with 0 and increase to

5. The coefficients are symmetric.
To determine the expansion on
  we see
  thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of
  the pattern is as follows:
1. Introduce
  and then for each successive term reduce the exponent on
  by 1 until
  is reached.
2. Introduce
  and then increase the exponent on
  by 1 until
  is reached.

The next expansion would be
But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in . Pascal didn't invent the triangle. The underlying principles had been developed and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first published in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is named in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a number of probability problems.
To generate Pascal’s Triangle, we start by writing a 1. In the row below, row 2, we write two 1’s. In the 3rd row, flank the ends of the rows with 1’s, and add
  to find the middle number, 2. In the
  row, flank the ends of the row with 1’s. Each element in the triangle is the sum of the two elements immediately above it.
To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.

### Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such as
  can be a lengthy process.
Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.
Note the pattern of coefficients in the expansion of

The second term is
  The third term is
  We can generalize this result.

### Key Equations

### Key Concepts

1.
  is called a binomial coefficient and is equal to
  See .
2. The Binomial Theorem allows us to expand binomials without multiplying. See .
3. We can find a given term of a binomial expansion without fully expanding the binomial. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the binomial coefficient.
For the following exercises, use the Binomial Theorem to expand each binomial.
For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.
For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

### Graphical

For the following exercises, use the Binomial Theorem to expand the binomial
  Then find and graph each indicated sum on one set of axes.

### Extensions


# Sequences, Probability and Counting Theory

## Probability

### Learning Objectives

1. Introduction to Sample Spaces and Computing Basic Probabilities.

### Objective 1: Introduction to Sample Spaces and Computing Basic Probabilities.

Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.

The probability of an event is a description of how likely it is that an event will happen. A probability is a number between 0 and 1 (that is, between 0% and 100%), where probabilities closer to 100% are very likely to occur, and probabilities closer to 0% are very unlikely to occur. A probability of 0% means the event is impossible, and a probability of 100% means the event will certainly occur.

A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.

The sample space S for a probability model is the set of all possible outcomes. For example, the sample space for rolling a dice is the set 1,2,3,4,5,6.This notation is referred to as roster notation.

An event A is a subset of the sample space S. For example, the event “Rolling an even number” is the subset 2,4,6.

To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.

It is important to note that in order to use this formula, all outcomes must be equally likely to happen.

For example, the probability of rolling an even number with a standard dice is:

### Practice Makes Perfect

Residents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one in . They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line represents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name. In this section, we will investigate methods for making these types of predictions.

### Constructing Probability Models

Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is  An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event
 is a number that always satisfies
  where 0 indicates an impossible event and 1 indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like .
The sum of the probabilities listed in a probability model must equal 1, or 100%.

### Computing Probabilities of Equally Likely Outcomes

Let
 be a sample space for an experiment. When investigating probability, an event is any subset of
 When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in
 Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in
 so the probability of the event is

### Computing the Probability of the Union of Two Events

We are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card game, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability of the next card being a heart or a king. The union of two events
  is the event that occurs if either or both events occur.
Suppose the spinner in  is spun. We want to find the probability of spinning orange or spinning a

There are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is
  There are a total of 6 sections, and 2 of them have a
 So the probability of spinning a
 is
  If we added these two probabilities, we would be counting the sector that is both orange and a
 twice. To find the probability of spinning an orange or a
 we need to subtract the probability that the sector is both orange and has a

The probability of spinning orange or a
 is

### Computing the Probability of Mutually Exclusive Events

Suppose the spinner in  is spun again, but this time we are interested in the probability of spinning an orange or a
 There are no sectors that are both orange and contain a
 so these two events have no outcomes in common. Events are said to be mutually exclusive events when they have no outcomes in common. Because there is no overlap, there is nothing to subtract, so the general formula is
Notice that with mutually exclusive events, the intersection of
 and
 is the empty set. The probability of spinning an orange is
 and the probability of spinning a
 is
 We can find the probability of spinning an orange or a
 simply by adding the two probabilities.
The probability of spinning an orange or a
 is

### Using the Complement Rule to Compute Probabilities

We have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding the probability that an event will not happen. The complement of an event
 denoted
  is the set of outcomes in the sample space that are not in
 For example, suppose we are interested in the probability that a horse will lose a race. If event
 is the horse winning the race, then the complement of event
 is the horse losing the race.
To find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a probability model must be 1.
The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is
 the probability of the horse losing the race is simply

### Computing Probability Using Counting Theory

Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sample spaces. These problems can be complicated, but they can be made easier by breaking them down into smaller counting problems.
Assume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the probability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There are 5 phones that are not defective, so there are
 ways to select 2 phones that are not defective. There are 8 phones, so there are
 ways to select 2 phones. The probability of selecting 2 phones that are not defective is:

### Key Equations

### Key Concepts

1. Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
2. The probabilities in a probability model must sum to 1. See .
3. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See .
4. To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See .
5. To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See .
6. The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See .
7. In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See .

### Section Exercises

### Verbal

### Numeric

For the following exercises, use the spinner shown in  to find the probabilities indicated.
For the following exercises, two coins are tossed.
For the following exercises, four coins are tossed.
For the following exercises, one card is drawn from a standard deck of
  cards. Find the probability of drawing the following:
For the following exercises, two dice are rolled, and the results are summed.
For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:
For the following exercises, use this scenario: a bag of M&Ms contains
  blue,
  brown,
  orange,
  yellow,
  red, and
  green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

### Extensions

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting
  numbers from the numbers
  to
  After the player makes his selections,
  winning numbers are randomly selected from numbers
  to
  A win occurs if the player has correctly selected
  or
  of the
  winning numbers. (Round all answers to the nearest hundredth of a percent.)

### Real-World Applications

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov

### Chapter Review Exercises

### Sequences and Their Notation

### Arithmetic Sequences

### Geometric Sequences

### Series and Their Notation

### Counting Principles

### Binomial Theorem

### Probability

For the following exercises, assume two die are rolled.
For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

### Practice Test

For the following exercises, use the spinner in .


# Introduction to Calculus

## Introduction to Calculus

Sifan Hassan, an Ethiopian-born Dutch runner, has dominated distance running for several years. She became the first runner to win both the 1500 and 10,000 meter races at a World Championship. During the Tokyo Olympics, she joined only one other runner in history when she medaled in the rarely attempted distance triple: Winning the and won the gold medal in both the 5000 and 10,000 meter races and winning he bronze in the 1500. Hassan's signature racing style is to stay at the back of the pack for much of the race, and then move up during the final laps.
Hassan does not run at her top speed at every instant. How then, do we approximate her speed at any given instant? We will find the answer to this and many related questions in this chapter.


# Introduction to Calculus

## Finding Limits: Numerical and Graphical Approaches

Intuitively, we know what a limit is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in.https://en.wikipedia.org/wiki/Human_height and http://en.wikipedia.org/wiki/List_of_tallest_people Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was.
To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits.

### Understanding Limit Notation

We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence
gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function
  then as the input
  gets closer and closer to
  the output y-coordinate gets closer and closer to
  We say that the output “approaches”

 provides a visual representation of the mathematical concept of limit. As the input value
  approaches
  the output value
  approaches

We write the equation of a limit as
This notation indicates that as
  approaches
  both from the left of
  and the right of
  the output value approaches

Consider the function
We can factor the function as shown.
Notice that
  cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition,
  for the simplified function. We can represent the function graphically as shown in .
What happens at
  is completely different from what happens at points close to
  on either side. The notation
indicates that as the input
  approaches 7 from either the left or the right, the output approaches 8. The output can get as close to 8 as we like if the input is sufficiently near 7.
What happens at
  When
  there is no corresponding output. We write this as
This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as
Notice that the limit of a function can exist even when
  is not defined at
  Much of our subsequent work will be determining limits of functions as
  nears
  even though the output at
  does not exist.

### Understanding Left-Hand Limits and Right-Hand Limits

We can approach the input of a function from either side of a value—from the left or the right.  shows the values of
as described earlier and depicted in .
Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in  are

  and
  The corresponding outputs are
  and
  These values are getting closer to 8. The limit of values of
  as
  approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function
  as
  approaches 7.
Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in  are

  and
  The corresponding outputs are

  and
  These values are getting closer to 8. The limit of values of
  as
  approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function
  as
  approaches 7.
 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input
  within the interval
  to produce an output value of
  within the interval

We also see that we can get output values of
  successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.
 provides a visual representation of the left- and right-hand limits of the function. From the graph of
  we observe the output can get infinitesimally close to
  as
  approaches 7 from the left and as
  approaches 7 from the right.
To indicate the left-hand limit, we write
To indicate the right-hand limit, we write

### Understanding Two-Sided Limits

In the previous example, the left-hand limit and right-hand limit as
  approaches
  are equal. If the left- and right-hand limits are equal, we say that the function
  has a two-sided limit as
  approaches
  More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

### Finding a Limit Using a Graph

To visually determine if a limit exists as
  approaches
  we observe the graph of the function when
  is very near to
  In  we observe the behavior of the graph on both sides of

To determine if a left-hand limit exists, we observe the branch of the graph to the left of
  but near
  This is where
  We see that the outputs are getting close to some real number
  so there is a left-hand limit.
To determine if a right-hand limit exists, observe the branch of the graph to the right of
  but near
  This is where
  We see that the outputs are getting close to some real number
  so there is a right-hand limit.
If the left-hand limit and the right-hand limit are the same, as they are in , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.
Finally, we can look for an output value for the function
  when the input value
  is equal to
  The coordinate pair of the point would be
  If such a point exists, then
  has a value. If the point does not exist, as in , then we say that
  does not exist.

### Finding a Limit Using a Table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of
  approach
  from both sides. Then we determine if the output values get closer and closer to some real value, the limit

Let’s consider an example using the following function:
To create the table, we evaluate the function at values close to
  We use some input values less than 5 and some values greater than 5 as in . The table values show that when
  but nearing 5, the corresponding output gets close to 75. When
  but nearing 5, the corresponding output also gets close to 75.
Because
then
Remember that
  does not exist.

### Key Concepts

1. A function has a limit if the output values approach some value
  as the input values approach some quantity
  See .
2. A shorthand notation is used to describe the limit of a function according to the form
  which indicates that as
  approaches
  both from the left of
  and the right of
  the output value gets close to

3. A function has a left-hand limit if
  approaches
  as
  approaches
  where
  A function has a right-hand limit if
  approaches
  as
  approaches
  where

4. A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
5. A graph provides a visual method of determining the limit of a function.
6. If the function has a limit as
  approaches
  the branches of the graph will approach the same
  coordinate near
  from the left and the right. See .
7. A table can be used to determine if a function has a limit. The table should show input values that approach
  from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See .
8. A graphing utility can also be used to find a limit. See .

### Section Exercises

### Verbal

### Graphical

For the following exercises, estimate the functional values and the limits from the graph of the function
  provided in .
For the following exercises, draw the graph of a function from the functional values and limits provided.
For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as
  approaches 0.
For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as
  approaches
  If the function has a limit as
  approaches
  state it. If not, discuss why there is no limit.

### Numeric

For the following exercises, use numerical evidence to determine whether the limit exists at
  If not, describe the behavior of the graph of the function near
  Round answers to two decimal places.
For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as
  approaches the given value.
For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as
  approaches
  If the function has a limit as
  approaches
  state it. If not, discuss why there is no limit.

### Extensions


# Introduction to Calculus

## Finding Limits: Properties of Limits

Consider the rational function
The function can be factored as follows:
Does this mean the function
  is the same as the function

The answer is no. Function
  does not have
  in its domain, but
  does. Graphically, we observe there is a hole in the graph of
  at
  as shown in  and no such hole in the graph of
  as shown in .
So, do these two different functions also have different limits as
  approaches 7?
Not necessarily. Remember, in determining a limit of a function as
  approaches
  what matters is whether the output approaches a real number as we get close to
  The existence of a limit does not depend on what happens when
  equals

Look again at  and . Notice that in both graphs, as
  approaches 7, the output values approach 8. This means
Remember that when determining a limit, the concern is what occurs near
  not at
  In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

### Finding the Limit of a Sum, a Difference, and a Product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.
Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

### Finding the Limit of a Polynomial

Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit of a polynomial function as
  approaches
  is equivalent to simply evaluating the function for
 .

### Finding the Limit of a Power or a Root

When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.

### Finding the Limit of a Quotient

Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.

### Key Concepts

1. The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See .
2. The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See  and .
3. The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See .
4. The limit of the root of a function equals the corresponding root of the limit of the function.
5. One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See .
6. Another method of finding the limit of a complex fraction is to find the LCD. See .
7. A limit containing a function containing a root may be evaluated using a conjugate. See .
8. The limits of some functions expressed as quotients can be found by factoring. See .
9. One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the limits algebraically.
For the following exercise, use the given information to evaluate the limits:

 .
For the following exercises, evaluate the following limits.
For the following exercises, find the average rate of change

### Graphical

For the following exercises, refer to .

### Real-World Applications


# Introduction to Calculus

## Continuity

Arizona is known for its dry heat. On a particular day, the temperature might rise as high as
  and drop down only to a brisk
   shows the function
  where the output of
  is the temperature in Fahrenheit degrees and the input
  is the time of day, using a 24-hour clock on a particular summer day.
When we analyze this graph, we notice a specific characteristic. There are no breaks in the graph. We could trace the graph without picking up our pencil. This single observation tells us a great deal about the function. In this section, we will investigate functions with and without breaks.

### Determining Whether a Function Is Continuous at a Number

Let’s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in  indicates that, at 2 a.m., the temperature was
 . By 2 p.m. the temperature had risen to
  and by 4 p.m. it was
  Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly
  In fact, any temperature between
  and
  occurred at some point that day. This means all real numbers in the output between
  and
  are generated at some point by the function according to the intermediate value theorem,
Look again at . There are no breaks in the function’s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a continuous function. Temperature as a function of time is an example of a continuous function.
If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let’s create the function
  where
  is the output representing cost in dollars for parking
  number of hours. See .
Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $25 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function’s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.
A function that remains level for an interval and then jumps instantaneously to a higher value is called a stepwise function. This function is an example.
A function that has any hole or break in its graph is known as a discontinuous function. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function.
So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let’s use the function
  represented in  as an example.
Condition 1 According to Condition 1, the function
  defined at
  must exist. In other words, there is a y-coordinate at
  as in .
Condition 2 According to Condition 2, at
  the limit, written
  must exist. This means that at
  the left-hand limit must equal the right-hand limit. Notice as the graph of
  in  approaches
  from the left and right, the same y-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at
 .
Condition 3 According to Condition 3, the corresponding
  coordinate at
  fills in the hole in the graph of
  This is written

Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in  so the function is continuous as

 through  provide several examples of graphs of functions that are not continuous at
  and the condition or conditions that fail.

### Identifying a Jump Discontinuity

Discontinuity can occur in different ways. We saw in the previous section that a function could have a left-hand limit and a right-hand limit even if they are not equal. If the left- and right-hand limits exist but are different, the graph “jumps” at
 . The function is said to have a jump discontinuity.
As an example, look at the graph of the function
  in . Notice as
  approaches
  how the output approaches different values from the left and from the right.

### Identifying Removable Discontinuity

Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let’s look at the function
  represented by the graph in . The function has a limit. However, there is a hole at
 . The hole can be filled by extending the domain to include the input
  and defining the corresponding output of the function at that value as the limit of the function at
 .

### Recognizing Continuous and Discontinuous Real-Number Functions

Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of
  as
  approaches
  is the same as the value of
  when
  So
  There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.

### Determining the Input Values for Which a Function Is Discontinuous

Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up.
To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.

### Determining Whether a Function Is Continuous

To determine whether a piecewise function is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.

### Key Concepts

1. A continuous function can be represented by a graph without holes or breaks.
2. A function whose graph has holes is a discontinuous function.
3. A function is continuous at a particular number if three conditions are met:

4. A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to “jump.”
5. A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous. See .
6. Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain. See  and .
7. For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine why the function
  is discontinuous at a given point
  on the graph. State which condition fails.
For the following exercises, determine whether or not the given function
  is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

### Graphical

For the following exercises, refer to . Each square represents one square unit. For each value of
  determine which of the three conditions of continuity are satisfied at
  and which are not.
For the following exercises, use a graphing utility to graph the function
  as in . Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.
For the following exercises, consider the function shown in .