We can express these sequences using an explicit equation which describes the nth term of an arithmetic sequence in terms of the first term and the common difference.

\displaystyle a\left(n\right)=a_1+d\left(n-1\right)

\bm{a\left(n\right)}

the nth term in the sequence

\bm{a_1}

the first term in the sequence

\bm{d}

the common difference between terms

An arithmetic sequence is an example of a function that has a constant rate of change, which we can call a linear function. Linear functions can be written in the form:

\displaystyle f\left(x\right) = mx + b

\bm{m}

the slope of the line

\bm{b}

the y-value of the y-intercept

The graph of a linear function is a straight line.

In general, any relationship which has a constant rate of change is a linear relationship. A relationship which does not have a constant rate of change is called a nonlinear relationship.

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A linear relationship

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A nonlinear relationship

A linear relationship can be defined by the slope of the line and any point on the line.

For this reason, we can determine the equation of linear function from a graph, table, description, or even just two points. All we need to do is find the rate of change for the value of m, then substitute a point on the line into the equation form to solve for the value of b.

Worked examples

Example 1

Emanuel is selling raffle tickets to raise money for charity. The table below shows the cumulative number of tickets he has sold each hour for the first three hours:

Time (hours)	1	2	3
Total ticket sales	14	28	42

State whether Emanuel's ticket sales represent a linear or nonlinear function.

Approach

A linear function will have a constant rate of change. We can compare the values in the table and see how much the total ticket sales are increasing by each hour.

Solution

Emanuel sells 14 tickets in the first hour. He then sells 28 - 14 = 14 tickets in the second hour, and 42 - 28 = 14 tickets in the third hour.

So the rate of change is constant and therefore the ticket sales represent a linear function.

Determine the rule which relates Emanuel's ticket sales and time.

Approach

The rule is what happens to one term to get to the next.

Solution

From part (a) we know that Emanuel is able to sell 14 additional raffle tickets each hour.

If Emanuel's ticket sales continue in this way, determine the total number of tickets he will have sold after 6 hours.

Solution

From part (b) we know that Emanuel is selling 14 tickets per hour. So after 6 hours, if the pattern stays the same, he will have sold 14 \cdot 6 = 84 raffle tickets.

Reflection

We can expand the table to see all of Emmanuel's ticket sales up to and including the 6th hour:

Time (hours)	1	2	3	4	5	6
Total ticket sales	14	28	42	56	70	84

Example 2

Tiles were stacked in a pattern as shown:

An image showing tiles stacked in a pattern with increasing stack height and corresponding number of tiles. Stack 1 has 1 tile. Stack 2 has 3 tiles. Stack 3 has 5 tiles. And stack 4 has 7 tiles

A table of values representing the relationship between the height of the stack and the number of tiles was partially completed.

Height of stack	1	2	3	4	5	10	100
Number of tiles	1	3

Describe the recursive pattern and write the explicit equation.

Approach

The recursive pattern will describe the change in the number of tiles going from each stack to the next.

The explicit equation is a function which can be used to find the number of tiles for a stack of any height.

Solution

The number of tiles from one stack to the next increases by 2. This tells us that the common difference is +2. We can also see that the first stack has one tile. So the explicit equation for this pattern is a\left(n\right)=1+2\left(n-1\right)where a\left(n\right) is the number of tiles and n is the height of the stack.

Complete the table of values representing the relationship between the height of the stack and the number of tiles.

Solution

We can use the pattern identified in part (a) to help us fill in the table.

Height of stack	1	2	3	4	5	10	100
Number of tiles	1	3	5	7	9	19	199

Reflection

We can double-check that the pattern found in part (a) is correct by comparing some of the results in the table to the images of the sequence of stacks.

Write a linear function relating the height of the stack, x, to the number of tiles in the stack, f\left(x\right).

Approach

We already have the explicit equation, which uses a\left(n\right) for the number of tiles and n for the stack height.

To find the linear function, we can replace n with x, and a\left(n\right) with f\left(x\right). After doing this, we can simplify the equation so that it is in the form f\left(x\right)=mx+b.

Solution

\displaystyle a\left(n\right)	\displaystyle =	\displaystyle 1+2\left(n-1\right)	Explicit equation
\displaystyle f\left(x\right)	\displaystyle =	\displaystyle 1+2\left(x-1\right)	Replace a\left(n\right) with f\left(x\right), and n with x
\displaystyle f\left(x\right)	\displaystyle =	\displaystyle 1+2x-2	Distributive property
\displaystyle f\left(x\right)	\displaystyle =	\displaystyle 2x-1	Simplify

Reflection

Another method for finding the linear function is to use the information available.

We know from part (a) that the common difference is 2, so we know that m=2 for the slope of the line.

We can then substitute in any point on the line (for example, x=1,\,f\left(x\right)=1) and solve for the value of b.

\displaystyle f\left(x\right)	\displaystyle =	\displaystyle 2x+b	Linear function form
\displaystyle 1	\displaystyle =	\displaystyle 2(1)+b	Substitute in x=1,\,f\left(x\right)=1
\displaystyle 1	\displaystyle =	\displaystyle 2+b	Evaluate the multiplication
\displaystyle -1	\displaystyle =	\displaystyle b	Subtraction property of equality

So the linear function will be f\left(x\right)=2x-1.

Example 3

Write the linear function for the graph:

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Approach

The linear function will be of the form y=mx+b where m is the slope and b is the y-value of the y-intercept of the graph.

We can find m as \dfrac{\text{rise}}{\text{run}}, and we can find b as the value where the line crosses the y-axis.

We can identify the y-intercept and another point, \left(3,2\right) for example, to help us find this information.

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Solution

Looking at the graph, we can see that for each increase of 1 unit in the y-value, the value of x increases by 3 units. So m=\frac{1}{3}.

Looking at the graph again, we can see that the line crosses the y-axis at y=1. So b=1.

So the linear function for the graph is y=\frac{1}{3}x+1.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

M1.F.BF.A.1

Build a function that describes a relationship between two quantities.*

M1.F.BF.A.1.A

Determine steps for calculation, a recursive process, or an explicit expression from a context.

M1.F.LE.A.1.A

Know that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

M1.F.LE.A.1.B

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

M1.F.LE.A.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a table, a description of a relationship, or input-output pairs.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP6

Attend to precision.

M1.MP8

Look for and express regularity in repeated reasoning.

5.05 Linear patterns and sequences

Concept summary

Worked examples

Example 1

Approach

Solution

Approach

Solution

Solution

Reflection

Example 2

Approach

Solution

Solution

Reflection

Approach

Solution

Reflection

Example 3

Approach

Solution

Outcomes

M1.N.Q.A.1

M1.N.Q.A.1.C

M1.F.BF.A.1

M1.F.BF.A.1.A

M1.F.LE.A.1.A

M1.F.LE.A.1.B

M1.F.LE.A.2

M1.MP1

M1.MP2

M1.MP3

M1.MP6

M1.MP8

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