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When a quantity changes in a repetitive and specific manner, it exhibits a pattern. This behavior often happens in everyday life. Consider bacteria; they multiply over a particular period of time while following a specific pattern. Here, the bacteria is shown doubling.

Patterns are also very common in the process of making things. Consider a car factory that bases its production of cars following a specific pattern. In their case, they have planned for the number of cars they produce to increase by the same amount each month.

Patterns like the given scenarios can be modeled using sequences. This lesson demonstrates these sequences and explores a special type called an arithmetic sequence.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Patterns
Linear Equations
Solving Multi-Step Equations
Input and Output of a Function
Linear Function

Challenge

How Many Circles Are in Each Figure?

Interact with the following applet by using the options given and try to identify a pattern among the figures. What would Figure

4

look like?

Focusing on the number of circles in each figure, how many circles will Figure

20

have?

Discussion

Defining Sequences

A sequence is an ordered list of objects or elements called terms. The terms are often represented using a variable labeled with indices that specify the positions of the terms in the sequence.

Depending on the number of terms, a sequence can be finite or infinite. Since it is not possible to list all the elements in an infinite sequence, it is common to place three dots after a few terms. These three dots indicate that the sequence continues indefinitely based on a specific pattern.

The values of the terms of a sequence can repeat.

Sequence With Repeating Terms a_{1} - 1, a_{2} 0, a_{3} 1, a_{4} 0, a_{5} - 1, a_{6} 0, a_{7} 1, \dots

Sequences can have all sorts of patterns. The examples below use the same starting term but result in different sequences due to the differences in the patterns. Note that it is common to use letters like

a_{n},

b_{n},

c_{n},

and so on to represent distinct sequences.

Discussion

A Special Type of Sequences: Arithmetic Sequences

An arithmetic sequence is a sequence that has a constant difference between consecutive terms — that is, the difference between the first and the second term is the same as the difference between the second and the third term, and so on. This difference is called the common difference and is usually denoted with

d .

For example, consider the sequence of all even positive integers

Arithmetic sequence: 2, 4, 6, 8, 10... with a common difference of 2.

For this sequence, the common difference is

d = 2 .

It is important to note that the common difference can take on negative values as well. Consider the following arithmetic sequence where the values decrease.

Arithmetic sequence: 54, 51, 48, 45, 42, ... with common difference of -3.

This is an arithmetic sequence with a common difference of

- 3 .

Pop Quiz

Identifying Arithmetic Sequences

The following applet shows the firsts five terms of an infinite sequence. Analyze them carefully and determine whether or not the sequence is arithmetic.

Interactive applet showing different infinite sequences

Discussion

Explicit Rule of an Arithmetic Sequence

Every arithmetic sequence can be described by a linear function that is defined for the set of counting numbers. This function, referred to as the explicit rule of an arithmetic sequence, follows a specific general format.

$a_{n} = a_{1} + (n - 1) d$

Here,

a_{1}

is the first term and

d

is the common difference of the sequence. This function receives the position of a term,

n,

as an input and returns the value of the term in that position,

a_{n},

as an output.

Proof

Justification Based on Induction

Every arithmetic sequence has a common difference

d .

Therefore, it is possible to obtain every term of the sequence by adding the common difference to the first term

a_{1}

an appropriate number of times.

Tables can help in identifying the pattern and writing a general expression.

$n$	$a_{n}$	Rewrite
$1$	$a_{1}$	$a_{1} + 0 \cdot d$
$2$	$a_{2}$	$a_{1} + 1 \cdot d$
$3$	$a_{3}$	$a_{1} + 2 \cdot d$
$4$	$a_{4}$	$a_{1} + 3 \cdot d$
$5$	$a_{5}$	$a_{1} + 4 \cdot d$

The coefficient of the common difference is always $1$ less than the value of the position $n .$ This makes it possible to write an explicit rule like the following formula.

$a_{n} = a_{1} + (n - 1) d$

Proof

Proof by Using the Point-Slope Form of a Line

A sequence can be thought of as a set of coordinate pairs where the first coordinate is the position

n

and the second coordinate is the term value

a_{n} .

(1, a_{1}), (2, a_{2}), (3, a_{3}), \dots

As the position increases by

1,

the value of the term increases, or decreases, by a constant. Therefore, the rate of change between two consecutive coordinate pairs is constant and equal to

d .

That means an arithmetic sequence is a linear function with a slope

d .

Therefore, the explicit rule for the sequence can be written by substituting the coordinate pair

(1, a_{1})

into the point-slope form of a line.

Point - Slope Form y - y_{1} = m (x - x_{1}) Explicit Rule a_{n} - a_{1} = d (n - 1)

Finally, the explicit rule can be rewritten to the form given at the beginning of this proof.

a_{n} - a_{1} = d (n - 1) ⇕ a_{n} = a_{1} + (n - 1) d

Example

Counting the Number of Seats in a Movie Theater

Davontay goes to a cinema to see a new movie about a kid who runs like the wind. Waiting for the movie to start, he notices something very interesting about the theater itself.

a The theater has its seats organized in such a way that as the number of rows increase, the number of seats increase by a fixed amount. The first four rows have

8,

10,

12,

and

14

seats, respectively.

Which of the following explicit rules describes this situation and can predict the number of seats in each row?

b Predict how many seats there are in the

8^{th}

row?

c If the theater were to be expanded to fit

50

rows, how many seats would be in the

5 0^{th}

row?

Hint

a The general form of an explicit rule of an arithmetic sequence is

a_{n} = a_{1} + (n - 1) d .

b In the explicit rule describing the situation,

a_{n}

represents the number of seats in the

n^{th}

row.

c In the explicit rule the row number is represented by the term position

n .

Solution

a The increment in the number of seats from row to row is constant. Therefore, this situation can be modeled using an arithmetic sequence. Let the term position

n

represent the row number and the term value

a_{n}

the corresponding number of seats. The constant increase in number of seats is the common difference of the sequence.

It can be seen that the

common

difference

d

in this case is

2,

and the

first

term

a_{1}

8 .

Recall that the explicit rule of an arithmetic sequence is of the following form.

a_{n} = a_{1} + (n - 1) d

By substituting the corresponding values, the explicit rule can be found.

a_{n} = 8 + (n - 1) 2

b The number of seats in the

8^{th}

row can be found by using the expression found in Part A and substituting in

n = 8 .

a_{n} = 8 + (n - 1) 2

▼

Substitute

8

for

n

and evaluate

Substitute

$n = 8$

a_{8} = 8 + (8 - 1) 2

SubTerm

Subtract term

a_{8} = 8 + (7) 2

Multiply

a_{8} = 8 + 14

AddTerms

Add terms

a_{8} = 22

Therefore, there are

22

seats in the

8^{th}

row.

c Similarly as in Part B, the number of seats in the

5 0^{th}

row can be found by evaluating the explicit formula for

n = 50 .

a_{n} = 8 + (n - 1) 2

▼

Substitute

50

for

n

and evaluate

Substitute

$n = 50$

a_{50} = 8 + (50 - 1) 2

SubTerm

Subtract term

a_{50} = 8 + (49) 2

Multiply

a_{50} = 8 + 98

AddTerms

Add terms

a_{50} = 106

Therefore, there would be

106

seats in the

5 0^{th}

row.

Example

Calculating the Total Distance Ran and Making Predictions

Davontay, inspired by a movie about running, downloaded an app that keeps track of his total running distance. It even predicts his running goals. He went for a long run on his first day using it and has followed that same route ever since. One day, he opened the app, but the free trial period was over, and his data was locked!

Smartphone screen indicating a total distance of 60 km for day 23 and 70 km for day 27

a After careful thought, Davontay realizes that since he always runs the same path, the total distance must have increased by the same amount each day. This means that he can model the situation using an arithmetic sequence! Determine his missing running data and find an explicit rule to model this situation.

b Davontay has a personal goal of running

150

kilometers in total. If he keeps running the same path every day, how many days will he need to reach his goal?

Hint

a The total distance accumulated increases by a constant amount each day. That is the common difference of the arithmetic sequence.

b Use the explicit rule found in Part A and let

a_{n}

150

kilometers. This way, the corresponding

n -

value represents the number of the day in which that distance will be reached.

Solution

a Since the same distance was added every day to the data, the total distance reported each day should increase by a constant amount

d .

This will be the common difference of the arithmetic sequence. The day number will be represented by the term's position, and the total distance represented by the term's value.

It can be seen that in

4

days (day

23

to day

27

) the total distance increased by

10

kilometers (

60

70

kilometers). Since each day the total distance increased by

d,

this

10

kilometers increment should be equal to

4 d .

10 = 4 d \Leftrightarrow d = 2.5

Recall the explicit rule of an arithmetic sequence.

a_{n} = a_{1} + (n - 1) d

The common difference found earlier can now be substituted into the formula.

a_{n} = a_{1} + (n - 1) 2.5

Next, find

a_{1} .

This can be done by evaluating the explicit rule in place of one of the known terms. As a demonstration, since it is given that

a_{23} = 60,

the rule can be evaluated at

n = 23

and solved for

a_{1} .

a_{n} = a_{1} + (n - 1) 2.5

Substitute

$n = 23$

a_{23} = a_{1} + (23 - 1) 2.5

Substitute

$a_{23} = 60$

60 = a_{1} + (23 - 1) 2.5

▼

Solve for

a_{1}

SubTerm

Subtract term

60 = a_{1} + (22) 2.5

Multiply

60 = a_{1} + 55

SubEqn

$LHS - 55 = RHS - 55$

5 = a_{1}

RearrangeEqn

Rearrange equation

a_{1} = 5

Therefore, the first term

a_{1}

5 .

With this information, the explicit rule can be completed.

a_{n} = 5 + (n - 1) 2.5

b It is required to determine the numbers of day that Davontay will reach

150

kilometers in total. That can be done by using the explicit formula found in Part A.

a_{n} = 5 + (n - 1) 2.5

By substituting

150

for

a_{n}

an equation for

n

will be obtained. The solution to the equation is the number of the day that corresponds to this distance.

a_{n} = 5 + (n - 1) 2.5

Substitute

$a_{n} = 150$

150 = 5 + (n - 1) 2.5

▼

Solve for

n

SubEqn

$LHS - 5 = RHS - 5$

145 = (n - 1) 2.5

DivEqn

$LHS / 2.5 = RHS / 2.5$

58 = n - 1

AddEqn

$LHS + 1 = RHS + 1$

59 = n

RearrangeEqn

Rearrange equation

n = 59

Therefore, Davontay needs

59

days to reach his personal goal of running

150

kilometers in total. He did all of this without needing to pay for the pro version, thanks to the help of arithmetic sequences. Maybe now he can go run like the wind!

Pop Quiz

Determining the Explicit Rule of an Arithmetic Sequence

The following applet shows two of the first five terms of an infinite arithmetic sequence. Analyze the sequence carefully and determine which of the three options is the correct explicit rule of the sequence.

Interactive applet showing different infinite arithmetic sequences and explicit rules

Pop Quiz

Finding the $n^{th}$ Term of an Arithmetic Sequence

The following applet shows the first five terms of an infinite arithmetic sequence. Determine the explicit rule of the sequence and use it to find the indicated term.

Discussion

Describing Arithmetic Sequences Using a Recursive Rule

It has been shown how an explicit rule describes an arithmetic sequence. This representation uses a formula that receives the term position as the input and returns the term's value as the output. However, an arithmetic sequence can also be described by using a recursive rule.

Concept

Recursive Rule of an Arithmetic Sequence

A recursive rule of an arithmetic sequence is a pair comprised of a recursive equation telling how the term $a_{n}$ is related to its preceding term $a_{n - 1}$ and the initial term of the sequence $a_{1} .$

$a_{1}, a_{n} = a_{n - 1} + d$

In the equation above,

d

represents the common difference. The following applet gives an example recursive rule for an arithmetic sequence. It shows how the rule can be used to determine the first five terms of the sequence.

Interactive applet showing the use of the recursive rule for an arithmetic sequence

Note that the recursive equation alone describes all different arithmetic sequences that have the same common difference.

The recursive equation, an =an-1+2, describes both the arithmetic sequence 2,4,6,8,10,... as well as the arithmetic sequence -5,-3,-1,1,3,...

This example demonstrates the importance of why it is needed to specify the initial term

a_{1}

in the recursive rule, as it uniquely defines the specific arithmetic sequence.

Now it will be explained, step by step, how to write the recursive rule for an arithmetic sequence and how to use it to find an unknown term's value.

Method

Writing a Recursive Rule for an Arithmetic Sequence

The recursive rule of an arithmetic sequence gives the first term of the sequence and a recursive equation.

$a_{1}, a_{n} = a_{n - 1} + d$

Consider an example arithmetic sequence.

a_{1} 5, a_{2} 8, a_{3} 11, a_{4} 14, \dots

To write the recursive rule there are three steps to follow.

Find the Common Difference

The first step is to find the common difference

d .

To do this, calculate the difference between any two

consecutive

terms .

a_{1} 5, a_{2} 8, a_{3} 11, a_{4} 14, \dots 11 - 8 = 3 \Rightarrow d = 3

What to Do When No Consecutive Terms Are Known?

If the sequence was known to be arithmetic but no consecutive terms were known, the common difference could still be found. In general, it is enough to know two terms of the arithmetic sequence and their positions. For example, consider that only

a_{2}

and

a_{4}

were known.

a_{1} ?, a_{2} 8, a_{3} ?, a_{4} 14, \dots

Recall the general form of an arithmetic sequence.

Since each term increases by

d,

a_{2}

is equal to

a_{1} + d

and

a_{4}

is equal to

a_{1} + 3 d .

a_{4} = a_{1} + 3 d a_{2} = a_{1} + 3 d \Rightarrow 14 = a_{1} + 3 d \Rightarrow 48 = a_{1} + 3 d

By subtracting the corresponding sides of the resulting equations, a single equation, which can be solved for

d,

is obtained.

14 - 8 = a_{1} + 3 d - (a_{1} + d)

▼

Simplify

SubTerm

Subtract term

6 = a_{1} + 3 d - (a_{1} + d)

Distr

Distribute $- 1$

6 = a_{1} + 3 d - a_{1} - d

SubTerms

Subtract terms

6 = 2 d

▼

Solve for

d

DivEqn

$LHS / 2 = RHS / 2$

3 = d

RearrangeEqn

Rearrange equation

d = 3

Identify the First Term of the Sequence

To completely define an arithmetic sequence the first term should also be determined. Note that the recursive equation alone would describe any arithmetic sequence with common difference

d = 3 .

From the sequence it can be seen that

a_{1} = 5 .

a_{1} 5, a_{2} 8, a_{3} 11, a_{4} 14, \dots

What to Do When $a_{1}$ Is Not Known?

Reconsider the case where only

a_{2}

and

a_{4}

are known. Two equations were obtained.

1418 = a_{1} + 3 d = a_{1} + d

Since it is now known that

d = 3,

this value can be substituted in any of the equations to find

a_{1} .

This will be illustrated using the latter equation.

8 = a_{1} + 3 \Rightarrow a_{1} = 5

Write the Equation Rule

Finally, by putting together the previous results and substituting the common difference and the first term value, the recursive rule for the sequence can be written.

a_{1}, a_{n} = a_{n - 1} + d ⇓ a_{1} = 5, a_{n} = a_{n - 1} + 3

Example

Using Recursive Rules to Describe Patterns

Davontay finds himself looking at his phone app too often when running. He has an idea! He will make patterns using heavy rocks at each kilometer mark along his running route so he does not have to check his phone to know what point his on the run.

Different figures made using rocks on the ground

Davontay wants to make a fourth figure following the same pattern. Before setting out to collect more heavy rocks, he wants to model the pattern to save time and energy.

a Show that the situation can be modeled by using an arithmetic sequence and find a recursive rule for it.

b Calculate the number of rocks needed for the next figure.

Answer

a Proof: See solution.
Recursive Rule:

a_{1} = 5,

a_{n} = a_{n - 1} + 4

a_{4} = 17

Hint

a Recall that a sequence is arithmetic if it has a common difference.

b Note that the value of the term

a_{n}

is the number of rocks needed for the figure

n .

Evaluate the recursive rule found in Part A for the appropriate

n -

value.

Solution

a First, it will be shown that the described situation can be modeled using a sequence. Let

a_{n}

be the number of rocks used for the

n -

th figure.

Now, to prove that the sequence obtained this way is arithmetic, the difference between consecutive terms will be calculated.

Since the difference is constant, the sequence is arithmetic. Recall that the recursive equation for an arithmetic sequence is of the following form.

a_{1}, a_{n} = a_{n - 1} + d

In this equation,

d

represents the common difference of the sequence which was already found to be

d = 4 .

This value will now be substituted into the general recursive equation.

a_{n} = a_{n - 1} + d ⇓ a_{n} = a_{n - 1} + 4

To finish writing the recursive rule, it is needed to specify the first term of the sequence

a_{1} .

This can be identified in the previous diagram.

Using this information the explicit rule will now be completed.

a_{1} = 8, a_{n} = a_{n - 1} + 4

b The value of the next term

a_{4}

is the number of rocks needed for the next figure. To find this term's value, the recursive equation will be evaluated at

n = 4 .

a_{n} = a_{n - 1} + 4

▼

Substitute values and evaluate

Substitute

$n = 4$

a_{4} = a_{4 - 1} + 4

SubTerm

Subtract term

a_{4} = a_{3} + 4

Substitute

$a_{3} = 13$

a_{4} = 13 + 4

AddTerms

Add terms

a_{4} = 17

Davontay needs to collect

17

rocks for the next figure to place along his running route.

Example

Using Recursive and Explicit Rules to Predict Total Savings

Davontay is celebrating the completion of reaching his running goal. His family and friends throw him a party. They play a raffle game and he wins $$ 50!$ He decides not to spend this money, instead, he will save it and put it in a piggy bank.

Picture showing the gifts that Ramsha received

Davontay is planning to add

$ 5

every week. Thanks to the math he has practiced, he knows that the following recursive rule models his savings.

a_{1} = 50, a_{n} = a_{n - 1} + 5

In this expression,

a_{n}

represents the total savings after the

n^{th}

week.

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Catch-Up and Review

Proof

Proof

Hint

Solution

Hint

Solution

What to Do When No Consecutive Terms Are Known?

What to Do When a1​ Is Not Known?

Answer

Hint

Solution

What to Do When $a_{1}$ Is Not Known?