Topics
6. Linear and Exponential Models
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United States of America
Grades 9-12

6.02 Arithmetic sequences

Lesson
Practice

Introduction

We explored sequences and recursive notation in the last lesson. This lesson focuses on arithmetic sequences that have a recursive pattern of addition. We will explore how to identify arithmetic sequences and write them using recursive notation.

Arithmetic sequences

An arithmetic sequence is a recursive pattern of addition, where the same number is added to each subsequent term.

Arithmetic sequence

A sequence of numbers that increases or decreases by a constant amount

The constant is called a common difference and is usually denoted by d. The sequences will increase when d is positive or decrease when d is negative. Because the terms are increasing or decreasing by a constant amount, they will form a straight line when plotted on a graph.

An arithmetic sequence is represented in recursive notation by the formula:

\displaystyle a_n=a_{n-1}+d
\bm{a_n}
nth term
\bm{a_{n-1}}
previous term
\bm{d}
common difference

The domain of any arithmetic sequence is a subset of the integers. The domain can begin from any non-negative integer but will most often begin at 0 or 1. If the domain begins from zero, we will be given a_0=c. If the domain begins from one, we will be given a_1=c.

Examples

Example 1

Determine if the following sequences are arithmetic:

a

-\dfrac{1}{3},\,-1,\,-\dfrac{5}{3},\,-\dfrac{7}{3},\, -3, \ldots

Worked Solution
Create a strategy

A sequence is arithmetic if the terms share a common difference.

Apply the idea
\displaystyle -1-\left(-\dfrac{1}{3}\right)\displaystyle =\displaystyle -\dfrac{2}{3}
\displaystyle -\dfrac{5}{3}-\left(-1\right)\displaystyle =\displaystyle -\dfrac{2}{3}
\displaystyle -\dfrac{7}{3}-\left(-\dfrac{5}{3}\right)\displaystyle =\displaystyle -\dfrac{2}{3}
\displaystyle -3-\left(-\dfrac{7}{3}\right)\displaystyle =\displaystyle -\dfrac{2}{3}

The difference between each of the terms is the same, so this sequence is arithmetic.

Reflect and check

When plotted, this sequence forms a linear relationship. This is confirmation that it is an arithmetic sequence.

1
2
3
4
5
x
-3
-2
-1
y
b
n123456
T_n-3-203712
Worked Solution
Create a strategy

We will determine if this is arithmetic by find the difference between terms.

Apply the idea
\displaystyle -2-(-3)\displaystyle =\displaystyle 1
\displaystyle 0-(-2)\displaystyle =\displaystyle 2
\displaystyle 3-0\displaystyle =\displaystyle 3

The difference between terms is not the same, so the sequence is not arithmetic.

c
1
2
3
4
5
n
2
4
6
8
10
12
14
16
a_n
Worked Solution
Create a strategy

An arithmetic sequence will form a linear pattern or a straight line when graphed because there will be a constant rate of change.

Apply the idea

Since the points in this sequence form a curve, it is not arithmetic.

Reflect and check

We can validate our answer by checking the difference between each output and verifying that it is not constant.

\displaystyle 16-8\displaystyle =\displaystyle 8
\displaystyle 8-4\displaystyle =\displaystyle 4
\displaystyle 4-2\displaystyle =\displaystyle 2
\displaystyle 2-1\displaystyle =\displaystyle 1

The differences are not the same, so this confirms the sequence is not arithmetic.

Example 2

An arithmetic sequence is defined by T_{n}=T_{n-1}-\dfrac{1}{5} where T_1=4.

Determine the next 5 terms of the sequence.

Worked Solution
Create a strategy

We already know the first term, so we need to find the 2nd through 6th terms. We can do that by using the rule with n=2,\,3,\,4,\,5,\,6.

Apply the idea

T_2=T_1-\dfrac{1}{5}=4-\dfrac{1}{5}=\dfrac{19}{5}

T_3=T_2-\dfrac{1}{5}=\dfrac{19}{5}-\dfrac{1}{5}=\dfrac{18}{5}

T_4=T_3-\dfrac{1}{5}=\dfrac{18}{5}-\dfrac{1}{5}=\dfrac{17}{5}

T_5=T_4-\dfrac{1}{5}=\dfrac{17}{5}-\dfrac{1}{5}=\dfrac{16}{5}

T_6=T_5-\dfrac{1}{5}=\dfrac{16}{5}-\dfrac{1}{5}=3

The next 5 terms of the sequence are \dfrac{19}{5},\,\dfrac{18}{5},\,\dfrac{17}{5},\,\dfrac{16}{5},\,3.

Reflect and check

If we didn't use the formula, we could have noticed from the formula that the common difference is -\dfrac{1}{5}.That means we needed to subtract \dfrac{1}{5} from each term to get the next one.

Example 3

The first term of an arithmetic sequence is 2. The fifth term is 26.

a

Solve for d, the common difference of the sequence.

Worked Solution
Create a strategy

To get from the first term to the fifth term, we needed to add the common difference 4 times.

T_1+d+d+d+d=T_5

We can use this to set up an equation and solve for d.

Apply the idea
\displaystyle T_1+d+d+d+d\displaystyle =\displaystyle T_5Equation for fifth term
\displaystyle T_1+4d\displaystyle =\displaystyle T_5Combine like terms
\displaystyle 2+4d\displaystyle =\displaystyle 26Substitute T_1=2 and T_5=26
\displaystyle 4d\displaystyle =\displaystyle 24Subtract 2 from both sides
\displaystyle d\displaystyle =\displaystyle 6Divide both sides by 4

The common difference is 6.

b

Write a recursive rule for T_n in terms of T_{n-1} which defines this sequence and an initial term, T_1.

Worked Solution
Create a strategy

The first term was given to us initially: T_1=2. We can use the common difference we found in part (a) with the recursive formula for an arithmetic sequence: a_n=a_{n-1}+d.

Apply the idea

T_n=T_{n-1}+6 where T_1=2

Reflect and check

We can check our answer by using the rule to verify the 5th term is 26.

T_2=T_1+6=2+6=8

T_3=T_2+6=8+6=14

T_4=T_3+6=14+6=20

T_5=T_4+6=20+6=26

Example 4

Emmanuel 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)123
Total ticket sales142842
a

State whether Emmanuel's ticket sales represent an arithmetic sequence.

Worked Solution
Create a strategy

An arithmetic sequence 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.

Apply the idea

Emmanuel 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.

The rate of change is constant and therefore, the ticket sales represent an arithmetic sequence.

b

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

Worked Solution
Create a strategy

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

Apply the idea

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

a_n=a_{n-1}+14 where a_1=14

c

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

Worked Solution
Create a strategy

Although we could use the recursive rule to find the 6th term, we could also use the fact that Emmanuel is selling 14 tickets per hour.

Apply the idea

So after 6 hours, if the pattern stays the same, he will have sold 14 \cdot 6 = 84 raffle tickets.

Reflect and check

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

Time (hours)123456
Total ticket sales142842567084
Idea summary

The recursive formula for an arithmetic sequence is:

\displaystyle a_n=a_{n-1}+d
\bm{a_n}
nth term
\bm{a_{n-1}}
previous term
\bm{d}
common difference

Outcomes

F.BF.A.1

Write a function that describes a relationship between two quantities.

F.BF.A.1.A

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

F.BF.A.2

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

F.LE.A.1.B

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

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