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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.
### Catch-Up and Review

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*.

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

Challenge

Discussion

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 Termsa_{1}-1, a_{2}0, a_{3}1, a_{4}0, a_{5}-1, a_{6}0, a_{7}1, … $

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

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

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.

This is an arithmetic sequence with a common difference of $-3.$

Pop Quiz

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

Discussion

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.

$a_{n}=a_{1}+(n−1)d$

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 |
---|---|---|

$1$ | $a_{1}$ | $a_{1}+0⋅d$ |

$2$ | $a_{2}$ | $a_{1}+1⋅d$ |

$3$ | $a_{3}$ | $a_{1}+2⋅d$ |

$4$ | $a_{4}$ | $a_{1}+3⋅d$ |

$5$ | $a_{5}$ | $a_{1}+4⋅d$ |

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.

$a_{n}=a_{1}+(n−1)d$

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}),… $

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 Formy−y_{1}=m(x−x_{1})Explicit Rulea_{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

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.

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style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

b Predict how many seats there are in the $8_{th}$ row?

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c If the theater were to be expanded to fit $50$ rows, how many seats would be in the $50_{th}$ row?

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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 $n.$

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.

$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 $n=8.$
Therefore, there are $22$ seats in the $8_{th}$ row.

$a_{n}=8+(n−1)2$

▼

Substitute $8$ for $n$ and evaluate

Substitute

$n=8$

$a_{8}=8+(8−1)2$

SubTerm

Subtract term

$a_{8}=8+(7)2$

Multiply

Multiply

$a_{8}=8+14$

AddTerms

Add terms

$a_{8}=22$

c Similarly as in Part B, the number of seats in the $50_{th}$ 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

Multiply

$a_{50}=8+98$

AddTerms

Add terms

$a_{50}=106$

Example

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!

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.

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class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">3<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathdefault mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"><\/span><span class=\"mord\">4<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">n<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mord\">3<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

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?

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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 $a_{n}$ be $150$ kilometers. This way, the corresponding $n-$value represents the number of the day in which that distance will be reached.

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.

$10=4d⇔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

Multiply

$60=a_{1}+55$

SubEqn

$LHS−55=RHS−55$

$5=a_{1}$

RearrangeEqn

Rearrange equation

$a_{1}=5$

$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

$n=59$

Pop Quiz

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.

Pop Quiz

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

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

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 $a_{1}.$

$a_{1},a_{n}=a_{n−1}+d$

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

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

$a_{1},a_{n}=a_{n−1}+d$

$a_{1}5, a_{2}8, a_{3}11, a_{4}14, … $

To write the recursive rule there are three steps to follow.
1

Find the Common Difference

The first step is to find the common difference $d.$ To do this, calculate the difference between any two $consecutive$ $terms.$
### 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}5, a_{2}8, a_{3}11, a_{4}14, … 11−8=3⇒d=3 $

$a_{1}?, a_{2}8, a_{3}?, a_{4}14, … $

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 $a_{1}+3d.$
$a_{4}=a_{1}+3da_{2}=a_{1}+3d ⇒14=a_{1}+3d⇒48=a_{1}+3d $

By subtracting the corresponding sides of the resulting equations, a single equation, which can be solved for $d,$ is obtained.
$14−8=a_{1}+3d−(a_{1}+d)$

$6=2d$

▼

Solve for $d$

$d=3$

2

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 $a_{1}=5.$
### 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.

$a_{1}5, a_{2}8, a_{3}11, a_{4}14, … $

$1418 =a_{1}+3d=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⇒a_{1}=5 $

3

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

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.

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.

a **Proof:** See solution.

**Recursive Rule:** $a_{1}=5,$ $a_{n}=a_{n−1}+4$

b $a_{4}=17$

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.

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

$a_{4}=17$

Example

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.

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. <