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$n = 8$

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

Subtract term

a_{8} = 8 + (7) 2

a_{8} = 8 + 14

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

$n = 50$

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

Subtract term

a_{50} = 8 + (49) 2

a_{50} = 8 + 98

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

$n = 23$

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

$a_{23} = 60$

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

Solve for

a_{1}

Subtract term

60 = a_{1} + (22) 2.5

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

$a_{n} = 150$

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

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

$n = 4$

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

Subtract term

a_{4} = a_{3} + 4

$a_{3} = 13$

a_{4} = 13 + 4

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.

a Use the recursive formula to find the savings after each of the first four weeks.

b Davontay wants to know how much money he will save after one year, that is

52

weeks. There is one catch, he does not want to use the recursive formula more than

50

times! Help Davontay find the corresponding explicit rule and the savings after one year.

c Thinking bigger picture, Davontay is now thinking about opening a bank account to save his money. A bank agent suggests for him to deposit the money in an account set to the following explicit rule.

a_{n} = 100 + (n - 1) 10

In this equation

a_{n}

represents the total savings after the

n^{th}

week. Davontay would prefer to have a recursive rule since he thinks that it makes the weekly savings calculation easier. Help Davontay find the corresponding recursive rule.

Answer

a Savings After Week

1 :

$ 50

Savings After Week $2 :$ $$ 55$
Savings After Week $3 :$ $$ 60$

Savings After Week

4 :

$ 65

b Savings After One Year:

$ 305

c Recursive Rule:

a_{1} = 100,

a_{n} = a_{n - 1} + 10

Hint

a Note that the initial amount of

$ 50

represents the savings after week

1 .

The savings after week

2,

3,

and

4

can be found by evaluating the recursive rule at

n = 2,

n = 3,

and

n = 4,

respectively.

b Identify the relevant values from the recursive rule provided. Recall that the general form of a recursive rule is

a_{1},

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

where

d

is the common difference,

n

the position of the term and

a_{n}

the

n^{th}

term's value.

c Identify the relevant values from the recursive rule provided. Recall that the general form of a recursive rule is

a_{1},

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

where

d

is the common difference,

n

the position of the term and

a_{n}

the

n^{th}

term's value.

Solution

a Remember, it is already understood that

a_{1} = 50,

which means that the savings after the first week is

$ 50 .

Now, in order to find the savings for the second week, the recursive rule will be used for

n = 2 .

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

Substitute values and evaluate

$n = 2$

a_{2} = a_{2 - 1} + 5

Subtract term

a_{2} = a_{1} + 5

$a_{1} = 50$

a_{2} = 50 + 5

Add terms

a_{2} = 55

The total savings after week

2

$ 55 .

The same procedure can be repeated for

n = 3

and

n = 4 .

It is important to follow this procedure in order, since any term value can only be figured out if the previous term is known — just like the value of

a_{1}

was needed to find the value of

a_{2} .

The following table shows a summary of the calculations.

$a_{1} = 50, a_{n} = a_{n - 1} + 5$
Position	Substitute	Simplify
$n = 1$	$a_{1} = 50$	$a_{1} = 50$
$n = 2$	$a_{2} = a_{1} + 5 ⇓ a_{2} = 50 + 5$	$a_{2} = 55$
$n = 3$	$a_{3} = a_{2} + 5 ⇓ a_{3} = 55 + 5$	$a_{3} = 60$
$n = 4$	$a_{4} = a_{3} + 5 ⇓ a_{4} = 60 + 5$	$a_{4} = 65$

b The given recursive rule can compared with the general form of a recursive rule. This helps to identify the first term

a_{1}

and the common difference

d,

which are needed to write the explicit rule.

General Recursive Rule a_{1}, a_{n} = a_{n - 1} + d Given Recursive Rule a_{1} = 50, a_{n} = a_{n - 1} + 5

It is now understood that

a_{1} = 50

and

d = 5 .

Next, use the general form of an explicit rule and substitute the previously identified values.

a_{n} = a_{1} + (n - 1) d ⇓ a_{n} = 50 + (n - 1) 5

Finally to find the value after a years worth of savings — 52 weeks — the explicit rule will be evaluated using

n = 52 .

a_{n} = 50 + (n - 1) 5

Substitute

n

for

52

and evaluate

$n = 52$

a_{52} = 50 + (52 - 1) 5

Subtract term

a_{52} = 50 + (51) 5

a_{52} = 50 + 255

Add terms

a_{52} = 305

Davontay will save

$ 305

after one year.

c To help Davontay find the corresponding recursive rule, identify the first term

a_{1}

and the common difference

d

from the explicit rule given by the bank agent. This can be done by using the general form of an explicit rule.

General Explicit Rule a_{n} = a_{1} + (n - 1) d Given Explicit Rule a_{n} = 100 + (n - 1) 10

As seen,

a_{1} = 100

and

d = 10 .

Next, recall the general form of a recursive rule and substitute the values for

a_{1}

and

d .

a_{1}, a_{n} = a_{n - 1} + d ⇓ a_{1} = 100, a_{n} = a_{n - 1} + 10

Pop Quiz

Relating Recursive and Explicit Rules of Arithmetic Sequences

Find the corresponding explicit rule for the recursive rule shown.

Interactive applet showing different recursive and explicit rules

Discussion

Arithmetic Sequences and Linear Functions

It was previously shown that the terms' values of an arithmetic sequence change by a constant amount. Because of that consistent change, every arithmetic sequence has an associated linear relationship where the common difference is the slope of its associated line. To illustrate this, consider the following arithmetic sequence.

Arithmetic sequence: a_1 =6, a_2=10, a_3=12, a_4=14, a_5=16,...

The sequence has a common difference of

4 .

Its initial term is

a_{1} = 2 .

Using this information, its explicit rule can be stated.

a_{n} = a_{1} + (n - 1) d ↓ a_{n} = 2 + (n - 1) 4

This rule can be simplified by distributing and subtracting

4,

in that order.

a_{n} = 2 + (n - 1) 4

Distr

Distribute $4$

a_{n} = 2 + 4 n - 4

Subtract term

a_{n} = 4 n - 2

The sequence can be represented graphically using this expression. Treat the terms position

n

as the independent variable, and treat the terms value

a_{n}

as the dependent one. See how the graph of a sequence compares to that of the linear function

f (x) = 4 x - 2 .

Note how the common difference of the sequence relates to the slope of the line.

Interactive graph of the sequence terms and associated associated line

The relationship between

n

and

a_{n}

in an arithmetic sequence fits the definition of a linear function. Therefore, an arithmetic sequence is a linear function and its explicit rule is its function rule. That is why the resemblance of the explicit rule of an arithmetic sequence to the Slope-Intercept Form of a linear function is remarkable.

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

f (x) = m x + b

The only difference is that an arithmetic sequence has a domain and range which are discrete. In other words, the variables

n

and

a_{n}

can only have specific values. For this reason, their graphs are a series of isolated points, not a solid line.

Closure

Modeling a Pattern Using an Arithmetic Sequence

Now that sequences have been introduced and understood, the pattern presented at the beginning of the lesson can be modeled using a sequence. Then, the number of circles for Figure

20

can be calculated. Interact with the applet once more, predicting the next figure.

How many circles will Figure

20

have?

Hint

The pattern shown can be modeled by an arithmetic sequence. Find the explicit rule of the sequence.

Solution

The pattern in the applet can be modeled using a sequence where the values of theterms are the number of circles and the positions of the terms are the figure numbers.

Since the number of circles used increases by a constant amount, this sequence is arithmetic.

To find the number of circles appearing in Figure $20,$ find the explicit rule for this arithmetic sequence. The explicit rule of an arithmetic sequence has the following general form.

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

In this equation, $a_{1}$ is the first term of the sequence and $d$ is the common difference. Both can be identified from the previous diagram.

The initial term $a_{1} = 4$ and $d = 3 .$ These values can be substituted in the general explicit rule form.

a_{n} = a_{1} + (n - 1) d ⇓ a_{n} = 4 + (n - 1) 3

Finally, to find the number of circles in Figure $20,$ the explicit rule will be evaluated at $n = 20 .$

a_{n} = 4 + (n - 1) 3

Substitute

n

for

20

and evaluate

$n = 20$

a_{20} = 4 + (20 - 1) 3

Subtract term

a_{20} = 4 + (19) 3

a_{20} = 4 + 57